Let $f : \mathbb{Q} → \mathbb{Q}$ be a ring homomorphism. Show that f is the identity.  Im having trouble with this problem. I started considering two elements in $\mathbb{Q}$ and the definition of ring homomorphism:  Let $x, y \in \mathbb{Q}$ such that $x = \frac{m}{n}, y=\frac{m'}{n'}$with $m,m',n,n'\in \mathbb{Z}$ then:  $f(\frac{m}{n}*\frac{m'}{n'}) = f(\frac{m}{n})*f(\frac{m'}{n'})$   and   $f(\frac{m}{n}+\frac{m'}{n'}) = f(\frac{mn'+m'n}{nn'})=f(\frac{m}{n})+f(\frac{m'}{n'})$   Clearly, my goal is to show that:  $f(\frac{m}{n}*\frac{m'}{n'}) =\frac{m}{n}*\frac{m'}{n'}$  or   $f(\frac{m}{n}+\frac{m'}{n'}) =\frac{m}{n}+\frac{m'}{n'}$  I've been thinking for like an hour and still achieve nothing, any ideas/hints?
If $f: \mathbb Q\to \mathbb Q$ is a homomorphism, prove that $f(x)=0$ for all $x\in\mathbb Q$ or $f(x)=x$ for all $x$ in $\mathbb Q$.  I'm wondering if you can help me with this one?
Prove: If a sequence converges, then every subsequence converges to the same limit. I need some help understanding this proof:  Prove: If a sequence converges, then every subsequence converges to the same limit.  Proof:  Let $s_{n_k}$ denote a subsequence of $s_n$. Note that $n_k \geq k$ for all $k$. This easy to prove by induction: in fact, $n_1 \geq 1$ and $n_k \geq k$ implies $n_{k+1} &gt; n_k \geq k$ and hence $n_{k+1} \geq k+1$.  Let $\lim s_n = s$ and let $\epsilon &gt; 0$. There exists $N$ so that $n&gt;N$ implies $|s_n - s| &lt; \epsilon$. Now $k &gt; N \implies n_k &gt; N \implies |s_{n_k} - s| &lt; \epsilon$.   Therefore: $\lim_{k \to \infty} s_{n_k} = s$.   What is the intuition that each subsequence will converge to the same limit I do not understand the induction that claims $n_k \geq k$ Let's say I have the domain johnsmith.com and I want people to reach me via email, what is the best email username to use?  Some examples:   john@johnsmith.com - redundant john.smith@johnsmith.com - redundant and looks kinda stupid mail@johnsmith.com - makes sense, but looks very "technical" me@johnsmith.com root@johnsmith.com - for the geeks among us ;-)   Of course, technically I can simply say that I will be able to receive *@johnsmith.com, however, for a better user experience I still just want one email address for public presence (website, business cards, etc.).  Any suggestions? Is it possible to add a user and restrict him to run special programs? For example after this user logs in, he can only open Firefox to use Internet and no other programs can be run by this user. For example in a terminal, commands will not be accessible when you delete some environment variables like $HOME. But is it possible to avoid running programs in a graphical shell, like Gnome? If yes, how? How do you keep parents of floated elements from collapsing? Although elements like &lt;div&gt;s normally grow to fit their contents, using the float property can cause a startling problem for CSS newbies: If floated elements have non-floated parent elements, the parent will collapse. For example:   &lt;div&gt;   &lt;div style="float: left;"&gt;Div 1&lt;/div&gt;   &lt;div style="float: left;"&gt;Div 2&lt;/div&gt; &lt;/div&gt;    The parent div in this example will not expand to contain its floated children - it will appear to have height: 0. How do you solve this problem? I would like to create an exhaustive list of solutions here. If you're aware of cross-browser compatibility issues, please point them out. Solution 1 Float the parent.   &lt;div style="float: left;"&gt;   &lt;div style="float: left;"&gt;Div 1&lt;/div&gt;   &lt;div style="float: left;"&gt;Div 2&lt;/div&gt; &lt;/div&gt;    Pros: Semantic code. Cons: You may not always want the parent floated. Even if you do, do you float the parents' parent, and so on? Must you float every ancestor element? Solution 2 Give the parent an explicit height.   &lt;div style="height: 300px;"&gt;   &lt;div style="float: left;"&gt;Div 1&lt;/div&gt;   &lt;div style="float: left;"&gt;Div 2&lt;/div&gt; &lt;/div&gt;    Pros: Semantic code. Cons: Not flexible - if the content changes or the browser is resized, the layout will break. Solution 3 Append a &quot;spacer&quot; element inside the parent element, like this:   &lt;div&gt;   &lt;div style="float: left;"&gt;Div 1&lt;/div&gt;   &lt;div style="float: left;"&gt;Div 2&lt;/div&gt;   &lt;div class="spacer" style="clear: both;"&gt;&lt;/div&gt; &lt;/div&gt;    Pros: Straightforward to code. Cons: Not semantic; the spacer div exists only as a layout hack. Solution 4 Set parent to overflow: auto.   &lt;div style="overflow: auto;"&gt;   &lt;div style="float: left;"&gt;Div 1&lt;/div&gt;   &lt;div style="float: left;"&gt;Div 2&lt;/div&gt; &lt;/div&gt;    Pros: Doesn't require extra div. Cons: Seems like a hack - that's not the overflow property's stated purpose. Comments? Other suggestions? What is the largest positive $n$ for which $n^3+100$ is divisible by $n+10$  I tried to factorize $n^3+100$, but $100$ is not a perfect cube. I wish it were $1000$. Suppose I have a dataframe consisting of six time series. In this dataframe, some observations are missing, meaning at some timepoints all time series contain a NA-value. In R, one possible imputation package that can be used to impute time series data is Amelia. However, this package does not work for observations that are completely missing. Are there other ways to impute my data? For what it's worth, the amount of missing observations is less than 20% of all observations. How to make two people play at the same time? How to change the line spacing in my list of figures? I am using the book documentclass and have a list of figures. After  I noticed that there is a blank line to separate figures that are in different chapters, but not if they are within the same chapter. Since I changed the numbering of my figures to be {1, 2, 3, etc.} instead of {2.1, 2.2, 3.1, etc.} this looks silly and I'd rather have a blank line between each figure, no matter what.  How do I do that?  Similar: I am new to salesforce. I dont have coding background. Please let me know bst way to learn apex coding? How to add text to end of line when pattern is matched? inputs:  line1 with the PATTERN that contains ( )  line2 with the PATTERN that contains ( )  lineN with the PATTERN that contains ( )    outputs:  line1 with the PATTERN that contains ( ) ; line2 with the PATTERN that contains ( ) ; ... lineN with the PATTERN that contains ( ) ;   I tried this:  find . -name "test.txt" -print | xargs sed -i "/PATTERN/ s/$)/); /g"   but it didn't work. Multiple monitors, multiple workspaces? I connected my laptop to my LCD. Using the laptop display hotkey I can switch between different modes - only laptop monitor on, only LCD monitor on, both on and display the same view, both on and the desktop is stretched (divided).  But, I'm looking for something else - I would like to have two workspaces, one visible on the laptop monitor and the other on the LCD.  Is that possible? Is there a GUI IDE for programming python just like java has netbeans?  I wanted one where we could drag &amp; drop test boxes &amp; labels just as we do in swing components using netbeans or eclipse.  Or is there an add-on present in either netbeans or eclipse itself? Getting Started with Unit Testing Unit testing is, roughly speaking, testing bits of your code in isolation with test code. The immediate advantages that come to mind are:  Running the tests becomes automate-able and repeatable You can test at a much more granular level than point-and-click testing via a GUI    My question is, what are the current &quot;best practices&quot; in terms of tools as well as when and where to use unit testing as part of your daily coding? Lets try to be somewhat language agnostic and cover all the bases. I want to be able to close an alert box automatically using Javascript after a certain amount of time or on a specific event (i.e. onkeypress).  From my research, it doesn't look like that's possible with the built-in alert() function.  Is there a way to override it and have control over the dialog box that it opens?  Also, I don't want an override that shows a hidden div as the alert.  I need an actual dialog box. Canonical URL for a home page and trailing slashes Datasets constructed for a purpose similar to that of Anscombe's quartet I've just come across  (four datasets that have almost indistinguishable descriptive statistics but look very different when plotted) and I am curious if there are other more or less well-known datasets that have been created to demonstrate the importance of certain aspects of statistical analyses. I'd like to show that, given to hermitian operators $A,G$ on a Hilbert space $\mathscr{H}$, the following identity holds: $$ e^{iG\lambda}A e^{-iG\lambda} = A + i\lambda [G,A] + \frac{\left(i\lambda\right)^2}{2!}[G,[G,A]]+\ldots+\frac{(i\lambda)^n}{n!}\underbrace{[G,[G,[G,\ldots[G}_{n\ times},A]]]\ldots]+\ldots $$ where $\lambda$ denotes a real parameter and $[\ \!,]$ indicates the commutator.  This is a proof left to the reader by Sakurai in his books on Modern Quantum Mechanics. If the series $\sum_0^\infty a_n$ converges, then so does $\sum_1^\infty \frac{\sqrt{a_n}}{n} $ Problem:      Suppose that for every $n\in\mathbb{N}$, $a_n\in\mathbb{R}$ and $a_n\ge 0$. Given that   $$\sum_0^\infty a_n$$   converges, show that    $$\sum_1^\infty \frac{\sqrt{a_n}}{n} $$   converges.   Source: Rudin, Principles of Mathematical Analysis, Chapter 3, Exercise 7. Proof that the irrational numbers are uncountable If a ring element is right-invertible, but not left-invertible, then it has infinitely many right-inverses. Let $A$ be a ring and $a\in A$ an element that has a right-inverse but does not have a left-inverse. Show that $a$ has infinitely many right-inverses.