If $X$ is a normed linear space and $(x_n)$ is a Cauchy sequence in $X$, then $x_n$ has a subsequence $x_{n_k}$ which satisfies $$\|x_{n_k} - x_{n_{k-1}}\| &lt; \frac 1 {2^k}$$ for every $k &gt; 0$.  Why is this assertion true?
A sequence $\{x_j\}$ is said to be fast-Cauchy if $\sum_1^\infty d(x_j,x_{j+1})&lt;+\infty$. Show that every Cauchy sequence has a fast-Cauchy subsequence.   **My attempt:**Argue by contradiction, suppose there is no such subsequence. Then for any subsequence $\{{x_j}_k\}$,  $\sum_1^\infty d({x_j}_k,{x_j}_{k+1})$ diverges. Vaguely recall from an analysis course, if a series diverges, it doesn't not satisfy the Cauchy criterion. That is to say, there exists $\epsilon&gt;0$ such that for all $k$ and all $p\ge1$, $d({x_j}_{k+1},{x_j}_{k+2})+...+d({x_j}_{k+p}+{x_j}_{k+p+1})\ge\epsilon$.  Then I intuitively think that this is a contradiction, since if the original series is Cauchy, then after a large $N$, since $d(x_m,x_n)$ has to be very small, so that their sum should be smaller than given $\epsilon$, which is a contradiction. But I don't know how I can write down the concrete proof?
What does [param: NotNull] mean in C#? Industry standard file format deliverables for logos? Why do I have different IP addresses on different websites? My wireless router shows the IP address is 113.46.75.*, which some websites like ip138.com, www.ip.cn show as my IP.   But other websites, like googling for "what is my ip", www.whatismyip.com, www.whatsmyip.org, show that my IP is 121.18.126.*.   I am very curious why this happens and how is it possible.   So, why do different websites show that I have different IPs?  Thanks all!  Plus: I changed my IP address several times (restarting the modem) and all addresses is like 113.46.*.*, but my IP address does not change when googling for "what is my ip".  My ISP is a small company, so I think it is true that when visiting some websites, I am NATted or proxied to some IP addresses which have access to those websites -- not to avoid firewall though, since I am still blocked from some websites.  But I still want some more information or resources to support our guess. Thank you! Why I can't apply mirror modifier? Here's screen Why should we typedef a struct so often in C? Is there a closed formula for this summation? I have the summation  $$\sum\limits_{i=1}^n \frac{1}{i^2}$$  And I don't know how to find a closed formula for it. Any ideas? Is it possible to set start-up applications to initialize in certain workspaces? I'd like to not have to manually distribute my start-up applications across my numerous workspaces every time I boot up my machine. I'm running Ubuntu 12.04 .  Any Help ? I was thinking of how to create an IV for a block cipher that doesn't require stored state, and I came up with the idea of using an HMAC of the (padded) plaintext and a (possibly distinct) key as the IV.  This seems to be secure:   The IV will only be reused if there is an HMAC collision or if the plaintext and key are the same.  The former would imply that the HMAC is broken and the latter is harmless, since it is just producing the output twice. The HMAC is verified after decryption, but before padding is checked, so padding-oracle attacks should not be an issue.   In particular, if the HMAC and the block cipher use different keys, there seems to be a very simple proof of security, assuming that either (1) the HMAC cannot be distinguished from a random function or (2) the block cipher is secure against maliciously-chosen (but unique) IVs:   The attacker cannot forge the HMAC, since this would require knowledge of the signing key. Since the signing and encryption keys are different in this case, and the HMAC is assumed random, malicious IVs cannot be triggered by an attacker.   Alternatively, if the cipher is secure against malicious (but unique) IVs, then the security of the cipher+security of the HMAC seem to prove the security of the system.  This (obviously) provides no protection against replay attacks, but that requires state anyway.  However, I am not a cryptographer so I do not trust my conclusions on this matter.  If this technique is secure, it would appear to be quite useful in practice, as it is simple and provides both encryption and authentication. Why is $2\pi i \neq 0?$ We know that $e^{\pi i} = -1$ because of de Moivre's formula. ($e^{\pi i} = \cos \pi + i\sin \pi = -1).$  Suppose we square both sides and get $e^{2\pi i} = 1$(which you also get from de Moivre's formula), then shouldn't $2\pi i=0$? What am I missing here? Expected value of sums Suppose we draw cards out of a deck without replacement. How many cards do we expect to draw out before we get an ace? Has 'Ubuntu Software' been renamed to 'Snap Store'? I updated Ubuntu last night (sudo apt get update &amp;&amp; sudo apt get upgrade) and to  my surprise this morning I didn't find the 'Ubuntu Software' app and there's a new entry called 'Snap Store' which when opened looks just like the old 'Ubuntu Software' app. So has 'Ubunu Software' been renamed to 'Snap Store'? And if yes, why? Page Object Model or External property file for Selenium Object Repository I am a beginner in Selenium WebDriver (Java with TestNg). I'm trying to create a framework of my own.  My question here is what would be the better option to store my objects (basically all the webelements of my website under test), Page Object Model or External Property file? Why? Is there a better method than those? Please help. Show $\frac{1}{4}\leq \mu(A\cap B \cap C)$ Let $\{X,\mathcal{A},\mu\}$ be a probability space. Let $A$, $B$, $C$ be sets in $\mathcal{A}$, which satisfy $$\mu (A\cap B)\geq \frac{1}{2}, \quad \mu (A\cap C)\geq \frac{1}{2}, \quad \mu (B\cap C)\geq \frac{1}{2}.$$ Show that $\frac{1}{4}\leq \mu(A\cap B \cap C)$.  I've managed to show in a previous assignment that $$\mu(A\cap B \cap C)\leq \min\{\mu (A\cap B), \mu (A\cap C), \mu (B\cap C)\},$$ but now I'm stuck with the above inequality. I've tried to draw it, but still no luck. Any hints? Prove if $n$ has a primitive root, then it has exactly $\phi(\phi(n))$ of them Prove if $n$ has a primitive root, then it has exactly $\phi(\phi(n))$ of them.   Let $a$ be the primitive root then I know other primitive roots will be among $\{a,a^2,a^3 \cdots\cdots a^{\phi(n)} \}$ because any other number will be congruent modulo $n$ to one of these. Then I figured as the answer is $\phi(\phi(n))$ so for $a^k$ to be primitive root, $k$ must be coprime with $\phi(n)$.But I don't know the reason.Am I missing any silly point? What is an idiom/slang for "someone who pretends to be good when they're not"? This is not from real life, but from a movie on my local TV. A character in the movie is really bad, but when she talks with others, she pretends to be an innocent/ good woman. I want to know how to describe this kind of person. MySQL remove duplicates from big database quick Base64 Encode String in VBScript I have a web service load driver that's a Windows Script File (WSF), that includes some VBScript and JavaScript files. My web service requires that the incoming message is base64 encoded. I currently have a VBScript function that does this, but it's very inefficient (memory intensive, mostly due to VBScripts awful string concatenation)  [Aside; Yes, I've seen . The concatenation is happening in a loop across messages that are 1,000's to 10,000's bytes in size.]  I've tried using some custom string concatenation routines; one using an array and one using ADODB.Stream. These help, a little, but I think it would help more if I had some other way of encoding the message rather than via my own VBS function.  Is there some other way of encoding my message, preferebly using native Windows methods? I followed @Micha's instruction and successfully used the modules in GRASS. I use r.watershed to get the basin raster and the drainage line：  and I use r.water.outlet to get the watershed of one outlet：  I got the whole basin area of the green outlet point, but all I want is the little part of the original basin area, like this：part B is what I want  This schematic photo shows the idea of what I want：  Can I achieve this goal using r.water.outlet or other modules in GRASS？ Or all I can do is using the intersect tool in Geoprocessing tools in QGIS？  I want to find the watershed area, but using QGIS. I have following files to work with：  stream network shapefile(polyline) point shapefile clipped from the polyline shapefile a whole island(the stream is inside the island) DEM(I got 2 types:.lan &amp; .tiff)  I want to find the total watershed area located upstream of every point in the network, here's the schematic diagram：  the green, red, and yellow line enclose the watershed of each point(here I only take 3 points as the example), and I want to calculate their area. I already got the DEM file, and I must cut the region that contains the stream off, should I polygonize it first？ Can I directly use this form(.lan or .tiff) to work with GRASS？ Seeing close-vote details if I've already voted If I've already voted to close a question, it would still be nice to see the details of others' votes — the suggested migration if it's being marked as off topic, or the suggested duplicate if it's being marked as an exact duplicate, for instance. Instead, I only see "You have already voted to close this question" and the overall category. Keras - no prediction probability for multiple output models?