$a_n$ and $b_n$ are sequences of natural numbers, $\lim_{n-&gt;\infty}\frac{a_n}{b_n} = x$, where $x$ is irrational. Prove that $a_n$ and $b_n$ diverge to infinity.  I've proved that if a sequence of natural numbers converges to some number, then this sequence has to be constant for big $n$.
Let $\alpha$ be an irrational real number and let $a_j$ be a sequence of rational numbers converging to $\alpha$. Suppose that each $a_j$ is a fraction expressed in lowest terms: $a_j = \alpha_j / \beta_j$. Prove that the $\beta_j$ tend to $\infty$  Attempt: AFSOC that $\beta_j$ does not tend to infinity, then it is bounded by some $M$. We can also find an interval $\alpha \in (k, k+1)$. Let $\epsilon &gt; 0$, $\exists N_0$ such that for all $N &gt; N_0$ we have $|a_N - \alpha|  &lt; \epsilon$. So $|\frac{p_N}{q_N} - \alpha| &lt; \epsilon$. We try to use the fact that $q$ is bounded but fail to derive a contradiction. I do not know how to use the fact that $\alpha$ is between two integers, although it might not be relevant at all.  Ideas?
How should strace be used? A colleague once told me that the last option when everything has failed to debug on Linux was to use .  I tried to learn the science behind this strange tool, but I am not a system admin guru and I didn’t really get results.  So,   What is it exactly and what does it do? How and in which cases should it be used? How should the output be understood and processed?   In brief, in simple words, how does this stuff work? What is the best way to explain how to choose between "its" and "it's"? How do I get the Ender Dragon egg and what does it do? So I finally killed the Ender Dragon! As I was climbing the portal back to the world, I noticed that on top of the column there was an egg. So I attacked, mined, clicked and nothing happened, it just teleported away. What does the egg do and how do I get it? How do filament LED bulbs work, looking very similar to incandescent bulbs? Filament LED bulbs use very narrow strips containing the LED, which look similar to filaments of an old incandescent bulb:  They look like a normal bulb with large "old-style" coal filaments, and are of about the same size and shape as the common standart E27 socket bulb.    There are two basic variants of these bulbs:     One "plain" as described, looking impressively similar to a classic bulb The other has a white coating on the glass, about 15mm wide next to the metal socket.   How are these filaments build? I could imagine the it's "pretty normal" LEDs, but lots of them, directly bonded to something? (There are references to COB)  How does the heat management work? Ii have seen speculations about filling the bulb with either helium or "some patented organic gas" - could that be enough for cooling?  How does the driver - which seems to be very small - work?  In the variant with white coating - hiding some components, I assume - is already not that much space, and in the plain variant, there seems to be almost no space for something nontrivial.  Below you see some bulbs of the white coating variant, switched on:   (Images with the permission of this )    Note how the LED filament bulbs are not so much similar to the tungsten filament bulbs recently being phased out - they are more resembling the long outdated carbon filament bulbs: How does rooting work? Or, how do I root my device without a rooter?  Rooting instructions for Android devices that I have seen are typically of the following form:   Download Gingerbreak.apk Run Gingerbreak, click “root device” Observe that Superuser.apk has been magically installed, possibly along with busybox and other things, and enjoy your rooty goodness.   What I am wanting to know is this: how does the rooting actually happen? What does Gingerbreak do?  I understand the end result of rooting:   su is present and working Superuser.apk is installed and controls access to su Various utilities are present, provided by BusyBox (optional?)   I also think I understand the basic idea of how Gingerbreak gets started — it exploits some local privilege escalation vulnerability and uses its new-found powers to deploy Superuser and whatever else is necessary.  So, unless I am wrong in some of the understanding I laid out above, the core of thing I want to know is what the root installer does to deploy and activate Superuser. It seems that just putting Superuser.apk in the right place wouldn't be enough — how does it get allowed root access? How does it get hooked up to moderate that access for other applications?  What does Gingerbreak do post-exploit to enable and secure root access? List files sorted by the number of lines they contain "None is" or "none are" How does the Girl-Genius web-comic relate to the Agatha H. books? After spending some time reading  I was convinced to buy the book.  I've ordered what I think is the  in my local bookstore (yay local) and it seems to relate to the first one or two years of of web content, but I can't be sure.  My question is: "how does the book series line up with the Web comic?"  Does it contain specific episodes/adventures, relate to strips from a specific time-span or is it simply an assortment of episodes?  UPDATE: They are amazing.  It is interplanetary war in the introduction!!!!  Yes! I'm having a hard time finding the solution of the following equation:  $$0 = x\cos(x)+2$$  This is part of showing that $f(x):= x^2e^{\sin(x)}$ has 3 extremas in $[-\frac{\pi}{2},\frac{\pi}{2}]$. CNAME for top of domain? Is it possible to set a CNAME record at the top of a domain? (i.e. @ CNAME www, @ CNAME foobar.com., etc.)  My ISP says that it's only possible to use CNAME's for subdomains but I've read somewhere else that is should be possible even if not recommended. How to substantiate the claim that something is not known? I am writing a master's thesis (in Number Theory) and there are multiple places where I need to claim that something is not yet known. Since I am not an expert (and even if I were) I would like to reference some more authoritative source than myself or, say, Wikipedia.  For example, consider the following statement:     In 1980 Schmidt proved that [some property holds]. No other necessary or sufficient conditions are currently known, though.   For the first sentence I can cite Schmidt's original paper, but how can I substantiate the second claim? Rank of a Matrix Sum I have $3\times3$ matrices such that S=A+B. I know there is an inequality connecting rank of the matrices A, B and its sum S. Could you write down that here? It will be a great help for me.  Means equation  or inequality connecting rank(S), rank(A) and rank(B). It has been said that RSA uses a modulus product of two prime numbers for security reasons. But does RSA even work correctly if we allow composite integers instead?  I think that the answer is "NO".  Define “work correctly” as:   allows encryption and decryption to be carried out as in normal RSA, yielding the original message (message padding and unpadding can remain unspecified, or absent as in texbook RSA); a security level can be given in terms of the key generation process (which may have alternate requirements to testing that $p$ and $q$ are prime). This is a problem from Introduction to Representation Theory by Etingof et al. I have no idea how to approach it. My friend wrote a program that uses Gaussian elimination to find a basis, which starts with x, y, [x, y], [y, [x, y]], ... but freezes once it gets to elements with 9 or more occurrences of x, y.  I expect that there is some way to characterize the nonzero elements of this Lie algebra (i.e., elements that are "immune" to being shown to be zero through repeated application of the Jacobi identity) which could be turned into a way to enumerate them. I don't see what it is, though.  It's also totally unclear to me how to show that some set we know spans the Lie algebra (proven using some criterion as described in the previous paragraph) is actually linearly independent, unless we find a faithful representation... but I have no idea how to find a faithful representation for an infinite-dimensional Lie algebra.  Hints would be greatly appreciated.  Edit: After looking at the partial output, we can form the "straightforward" conjecture that:   If $n \geq 4$ is even, then there is one basis element of grade $n$, namely $\mathrm{ad}([x, y])^{(n-4)/2}[y, [y, [y, x]]]$. If $n \geq 5$ is odd, then there are two basis elements of grade $n$, namely $[x, z]$ and $[y, z]$ where $z$ is the single basis element of grade $n-1$.   I also have a different conjecture, which is mutually exclusive with the above: that if we flatten out all nonzero elements into the form ${\rm ad}(e_1) {\rm ad}(e_2) \ldots {\rm ad}(e_{n-1}) e_n$ using the Jacobi identity, then in the sequences $e_1, \ldots, e_{n-1}, e_n$ and $e_1, \ldots, e_n, e_{n-1}$, any span containing two x's must contain at least two y's and any span containing five y's must contain at least two x's, otherwise we can use the Jacobi identity to rearrange the element somehow to show that it vanishes. If this is true, then the long-range behaviour must be:   For $n \geq 8$ there is exactly one basis element of grade $n$, namely $[... [y, [x, [y, [y, [x, y]]]]] ...]$ (the pattern $y x y$ repeats).   I still don't see an approach to proving that none of the elements in this sequence actually vanishes, though.  Edit 2: Here is the hint given by Etingof in an email:     No, there are no published solutions, since these problems are used as   a homework for classes. This is indeed a nontrivial problem.      This is the positive part of the affine Kac-Moody algebra of type   A_2^2, which is the twisted loop algebra for sl(2). So what you need   to do is to relate this Lie algebra to loops into sl(2). I see from   your post that you are on the right track. Why is it that Frisian is considered the closest related language to English? It is commonly asked "What is the closest language to English?" and the equally common answer is         Except that there is rarely a reason given for this connection; the most that is given are baldly stated 'facts' (e.g. "English and Dutch are in the same 'Low German' group")  Can anyone give any substantive reasons? For example, an comparative overlap of vocabulary, a distance metric of phonology, privately shared syntactic rules among all the rules in West German languages. Are film capacitors affected by DC bias? After looking around, I have found quite a few things about how DC bias reduces effective capacitance for a ceramic, or electrolytic capacitor, however nothing seems to talk about the effect on film capacitors. I've found a couple of papers about film caps (such as: ), and they mention frequency and temperature, but I can't find anything about DC bias reducing capacitance.  I have obviously checked the data sheet of the component I plan on using (), but there isn't anything about DC bias in that (unless I've just failed at spotting it).  This suggests to me that they aren't affected by DC bias, but that seems a bit of a bold claim, and I can't see that stated anywhere.  Are film capacitor values affected by DC bias? If so, can anyone point me to some data about what the roll off is?  I am planning to use some film capacitors in an application with 48-60V DC across the capacitors and a ripple current around 30 amps. Do I need a visa for Paris, if my connecting flight stops for 24 hours in Paris? Do I need visa for Paris, if my connecting flight stops at Paris for 24hrs and I want to visit Paris during that 24hrs? I have US B1 visa. I'll be travelling from San Francisco to Bangalore (India) via Paris. I hold Indian passport. If yes, what is the procedure for that visa? Where can you ask about Career Advice? The  specifically says not to ask  for career advice but people do it anyway.  Should I consider the Help Center outdated or only loosely enforced?    And what about a career advice site on the network?  There are 70+ sites, and most of them are for relatively narrow interests (which is not a criticism, I enjoy looking through the many sections).  A career section seems like it would be very popular.  Also, a careers section would help keep career questions off the other sites. The startup disk cannot be partitioned or restored to a single partition. Windows 10 and Mojave The startup disk cannot be partitioned or restored to a single   partition.      The startup disk must be formatted as a single Mac OS Extended   (Journaled) volume or already partitioned by Boot Camp Assistant for   installing Windows.   I had installed Windows 10 via bootcamp in my mac. Current version is 10.14.1  Now I don't want windows and need that space back but getting above error when I go bootcamp to remove it.  This is the output of diskutil list  /dev/disk0 (internal, physical):    #:                       TYPE NAME                    SIZE       IDENTIFIER    0:      GUID_partition_scheme                        *251.0 GB   disk0    1:                        EFI EFI                     209.7 MB   disk0s1    2:                 Apple_APFS Container disk1         181.3 GB   disk0s2    3:       Microsoft Basic Data BOOTCAMP                60.3 GB    disk0s3    4:           Windows Recovery                         509.6 MB   disk0s4  /dev/disk1 (synthesized):    #:                       TYPE NAME                    SIZE       IDENTIFIER    0:      APFS Container Scheme -                      +181.3 GB   disk1                                  Physical Store disk0s2    1:                APFS Volume Macintosh HD            116.4 GB   disk1s1    2:                APFS Volume Preboot                 44.6 MB    disk1s2    3:                APFS Volume Recovery                512.8 MB   disk1s3    4:                APFS Volume VM                      4.3 GB     disk1s4   And this is of diskutil cs list  No CoreStorage logical volume groups found   Can anybody please help me to solve this problem?  Output after I tried commands given in the first answer  ❯ diskutil list /dev/disk0 (internal, physical):    #:                       TYPE NAME                    SIZE       IDENTIFIER    0:      GUID_partition_scheme                        *251.0 GB   disk0    1:                        EFI EFI                     209.7 MB   disk0s1    2:                 Apple_APFS Container disk1         250.8 GB   disk0s2  /dev/disk1 (synthesized):    #:                       TYPE NAME                    SIZE       IDENTIFIER    0:      APFS Container Scheme -                      +250.8 GB   disk1                                  Physical Store disk0s2    1:                APFS Volume Macintosh HD            116.4 GB   disk1s1    2:                APFS Volume Preboot                 44.6 MB    disk1s2    3:                APFS Volume Recovery                512.8 MB   disk1s3    4:                APFS Volume VM                      3.2 GB     disk1s4 What is object serialization?