If $(f_n):[0, 1] \to [0, 1]$ are continuous and converge to $f$ pointwise, must $f$ be Riemann Integrable?
Pointwise limit of continuous functions not Riemann integrable
How can I tell if a CVE has been fixed in Ubuntu's repositories? How to edit command line in full screen editor in ZSH? "debconf: DbDriver "config": config.dat is locked by another process: Resource temporarily unavailable" while installing packages Choosing C++ debugger on linux Can we have a way to edit bounty custom message? I put up  over on SO with the assumption that the text would be formatted like it is for a question or answer.  I'm not really sure if I was incorrect in my assumption, or if I just formatted my text improperly, as I can't go back and edit what I put in.  Am I missing an obvious feature?  If not, would it be reasonable to request bounty-text edits—perhaps with moderator-approval to prevent abuse?  Edit: I was trying to use markdown to produce an unordered list as part of my bounty.  What you see in the link is a blob of text with dashes separating phrases Modular arithmetic: division, fractions, solving linear congruences "Unanswered" question sort order on profile pages To help users keep track of (and hopefully get answers to) unanswered questions, it would be useful to have an "unanswered" sort order on a user's profile page.  It would order the list of questions with unanswered questions first.   This would be in addition to the "recent," "views," "newest," and "votes" sort orders. Reference Text that develops Linear Algebra with Knowledge of Abstract Algebra Background: Due to some unfortunate sequencing, I have developed my abstract algebra skills before most of my linear algebra skills. I've worked through Topics in Algebra by Herstein and generally liked his approach to vector spaces and modules. Besides a very elementary course in linear algebra (where most of the time went towards matrix multiplication), I have not developed any other linear algebra skills.  But it seems that it is now necessary for me to do so. Most of the topics that I am looking at now require some background of linear algebra and I still lack understanding of ideas like: bilinear forms, invariant subspaces, eigenvalues, requirements for diagonalization of a matrix and so forth.  This brings me to my question (provided there are any)     Question: What are some good texts that develop the theory of linear algebra from a perspective of general algebra?   Are there any texts that develop the key (elementary) ideas of linear algebra in an abstract setting? Thank you for the help! How many times to roll a die before getting two consecutive sixes? Basically, on average, how many times do you have to roll a fair six-sided die before getting two consecutive sixes? How to redirect stderr to a file "On their back" or "on their backs"? After the therapy, eight children (43%) became able to crawl/move on their back.   Or should I use "on their backs"?   Singular because each child only has one back, or plural because we're dealing with eight backs? The additive group $\mathbb{G}_a$ over an infinite field $k$ is not a topological group An algebraic group is not necessarily a topological group, since for an algebraic group, the product map $G \times G \to G$ is continuous when $G \times G$ is given the Zariski topology, which is finer than the product topology.   So does this tell you that the additive group $\mathbb{G}_a$ over an infinite field $k$ is not a topological group? The underlying variety of $\mathbb{G}_a \times \mathbb{G}_a$ is $\mathbb{A}_k^1 \times_k \mathbb{A}_k^1 = \mathbb{A}_k^2$, whose Zariski topology is the cofinite topology, where every open set is also open in the product topology (since $\mathbb{A}_k^1$ also has the Zariski topology). Should flour be thrown out past its expiry date? I want to put four figures in a single row. The code I am trying to use is taken from .  This puts three figures in a row. When I add a fourth one, I get three images in row and the last one appears in the second row. But I want all of them in one row. Here is my code.  \documentclass[12pt,twoside]{report}% Use this line for the print version of the thesis   \usepackage{graphicx} \usepackage{caption} \usepackage{subcaption} \begin{document}  \begin{figure}         \centering         \begin{subfigure}[b]{0.3\textwidth}                 \centering                 \includegraphics[width=\textwidth]{gull}                 \caption{A gull}                 \label{fig:gull}         \end{subfigure}%           \begin{subfigure}[b]{0.3\textwidth}                 \centering                 \includegraphics[width=\textwidth]{gull}                 \caption{A gull2}                 \label{fig:gull2}         \end{subfigure}%          ~ %add desired spacing between images, e. g. ~, \quad, \qquad etc.           %(or a blank line to force the subfigure onto a new line)         \begin{subfigure}[b]{0.3\textwidth}                 \centering                 \includegraphics[width=\textwidth]{tiger}                 \caption{A tiger}                 \label{fig:tiger}         \end{subfigure}         ~ %add desired spacing between images, e. g. ~, \quad, \qquad etc.           %(or a blank line to force the subfigure onto a new line)         \begin{subfigure}[b]{0.3\textwidth}                 \centering                 \includegraphics[width=\textwidth]{mouse}                 \caption{A mouse}                 \label{fig:mouse}         \end{subfigure}         \caption{Pictures of animals}\label{fig:animals} \end{figure} \end{document}   As I have mentioned above, it works for three figures.  But I can not use it for four figures. What should I change?   What does  width=\textwidth mean?   I changed    \begin{subfigure}[b]{0.3\textwidth}   to  \begin{subfigure}[b]{0.2\textwidth}   but it didn't help.  Then, I changed     \includegraphics[width=\textwidth]{mouse}   to  \includegraphics[width=0.5\textwidth]{mouse}   Still no solution. Getting a closed form from $\sum_{i=1}^{n-1} \sum_{j=i+1}^{n} \sum_{k=1}^{j} 1$ Display Tag Excerpt tooltips when editing a question Can I use B1 visa (Visitor-Business) to travel to the Canada without a business purpose? I got a B1 visa (Visitor-Business) for Canada in 2012 when I traveled to Canada for a business trip from my company. It expires in 2015 and I'd love to visit the Canada again, but for no other purpose than traveling, visiting family.  I want to know, If I can legally travel to Canada with B1 visa without a real business need (assuming I'm honest);  I live in India, and I have B1/B2 from US and used same Visa for Business &amp; Personal visits multiple times. Equivalence of conditions for Convergence of Non-negative Random Series Question  Let $X_n\geq 0$ be independent for $n\geq 1$. The following are equivalent.  $\sum_{n=1}^\infty X_n&lt;\infty$ a.s. $\sum_{n=1}^\infty[P(X_n&gt;1)+E(X_nI(X_n\leq 1)]&lt;\infty$ $\sum_{n=1}^\infty E(X_n/(1+X_n))&lt;\infty$   My attempt I was able to prove $(1)$ implies $(2)$ and $(2)$ implies $(3)$ but not $(3)$ implies $(1)$. For $(1)$ implies $(2)$, observe that by Borel-Cantelli $\sum_{n=1}^\infty P(X_n&gt;1)&lt;\infty$, else $X_n&gt;1$ infinitely often w.p.1 so that $(1)$ is violated. To see that $\sum_{n=1}^\infty E(X_nI(X_n\leq 1)]&lt;\infty$, apply Kolmogorov three-series theorem (I think). For $(2)$ implies $(3)$ we can write that $$ \begin{align} \frac{X_n}{1+X_n}&amp; =\frac{X_n}{1+X_n}I(X_n\leq 1)+\frac{X_n}{1+X_n}I(X_n&gt;1)\\ &amp;\leq X_nI(X_n\leq 1)+ I(X_n&gt;1). \end{align} $$ Now take the expectation of both sides and sum on $n$ using $(2)$ to conclude $(3)$. For the last implication $(3)$ implies $(1)$, I tried to use Kolmogorov three series theorem. To this end put $Y_n=X_nI(X_n\leq 1)$. First note that $\sum_{n=1}^\infty P(X_n&gt;1)&lt;\infty$ since $$ P(X_n&gt;1)\leq P\left(\frac{X_n}{1+X_n}&gt;\frac{1}{2}\right)\leq 2E\frac{X_n}{1+X_n} $$ by Markov's inequality. But I am unable to show that $\sum EY_n&lt;\infty$ and $\sum\text{Var}(Y_n)&lt;\infty$. Any help is appreciated. Difference between Hashing a Password and Encrypting it Recommendations for Numerical Analysis texts? I'm in a numerical analysis course right now and it's pretty rigorous but I'm enjoying it a lot. I took a lower level course before that was more oriented towards implementation of numerical methods, so it's not my first time with the material.  Anyway, my professor wrote his own course notes and they're fantastic but I wanted some reference material since I'm currently outlining my own notes. Just wondering what the big ones out there are  edit: I would say I'm at an advanced undergraduate level