In preparing for the quals, I stumbled across this question: Let $f$ be a continuous real-valued function on $[0,\infty)$ with $f(0) = 0$. Suppose that for each $y\in [0, 1]$ we have that $f(ny)\to 0$ as $n\to \infty$ through the integers. Prove that $f(x)\to 0$ as $x\to \infty$ through the reals. So, In trying to prove it I knew that if it were not the case then for some $\epsilon &gt; 0$ we have that there exist non-degenerate closed intervals, $A_n = [a_n, b_n]$ such that $a_n  \rightarrow \infty$ and $f(x) &gt; \epsilon$ for $x \in A_n.$ Perhaps then, I thought, an infinite number of $A_n$ non-trivially intersect, for a fixed $k$, $R_k = \{$Rationals with denominator $k\} $. If this were the case, then $f(n/k)$ would not got to 0 as $n \rightarrow \infty$. Unfortunately~! My attention was then brought to the counterexample of $A_k = [k + 1/k, k + 2/k]$... Darn. So, I guess that I need to use an irrational? I am not entirely sure... Any hints/insight would be most appreciated.
Can anyone help me with this problem?     Let $f:[0,\infty)\longrightarrow \mathbb R$ be a continuous function such that for each $x&gt;0$, we have $\lim_{n\to \infty}f(nx)=0$. Then prove that $\lim_{x\to \infty}f(x)=0$.   Our teacher told first to prove Baire's theorem, and then show that this is a consequence of that theorem. I proved Baire's theorem, and I spend a few hours thinking on how Baire's theorem is related to this problem, but I couldn't find anything. I'd really appreciate your help.
Best Place to put Feature Requests? How do I limit the number of rows returned by an Oracle query after ordering? Is there a way to make an Oracle query behave like it contains a MySQL limit clause? In MySQL, I can do this: select *  from sometable order by name limit 20,10  to get the 21st to the 30th rows (skip the first 20, give the next 10). The rows are selected after the order by, so it really starts on the 20th name alphabetically. In Oracle, the only thing people mention is the rownum pseudo-column, but it is evaluated before order by, which means this: select *  from sometable where rownum &lt;= 10 order by name  will return a random set of ten rows ordered by name, which is not usually what I want. It also doesn't allow for specifying an offset. Formatting a bootable USB that thinks it's a CD-ROM After upgrading to Windows 10, I remember that I had an old bootable recovery usb drive from the manufacturer that I no longer needed. I want to format it but it's showing up as a CD-ROM and format options are not available.  I also tried formatting it in a live linux mint environment, but linux cannot even mount the cd-rom.  How can I format this recovery usb that is showing up as a cd-rom? When clicking continue on the first Boot Camp screen,  I am presented with a dialog that displays:     The startup disk does not have enough space to be partitioned.   You must have at least 39 GB of free space available.     This is despite "About this Mac" reporting 271GB of free space:    And "Disk Utility" reporting 268GB of free space:    Searching online I found  which suggested it was the result of Time Machine which i recently enabled and suggests to run sudo tmutil thinlocalsnapshots / 999999999999 which I did, it ran sucessfully, and I restarted, yet to no avail.  What is the solution to this instance of Apple's poor engineering?  Running macOS High Sierra 10.13.4 (17E202) A uniformly continuous function maps bounded set to bounded sets I am trying to prove the following:  If $A\subset\mathbb R$ is bounded and $\,f:A\to \mathbb R\,$ is uniformly continuous, then $f[A]$ is bounded.  Could you check my proof?  Let $A \subseteq [-K,K]\subseteq \mathbb R$. Let $\varepsilon = 1$. If $f$ is uniformly continuous there is $\delta$ wuth $|x-y| &lt;\delta$ imply that $|f(x)-f(y)|&lt;1$ for all $x,y\in A$. Let $a \in A$. Then because $A$ is bounded there is a finite number of balls $B(a_n,\delta)$ that cover $A$. Let the number be $N$. Then $f(a)-N \le f(x) \le f(a) + N$ for all $x\in A$. How to remove version and thank you message from the admin footer I've used following codes and no luck  1  //Hide admin footer from admin function change_footer_admin () {     return ' '; } add_filter('admin_footer_text', 'change_footer_admin', 9999);  function change_footer_version() {     return ' '; } add_filter( 'update_footer', 'change_footer_version', 9999);   2  function wpbeginner_remove_version() { return ''; } add_filter('the_generator', 'wpbeginner_remove_version');   3  function my_footer_shh() {     remove_filter( 'update_footer', 'core_update_footer' );  } add_action( 'admin_menu', 'my_footer_shh' );   Still I see those texts and version on my footer.   So what is the accurate way to remove these? I've added these code snippets into my child theme function.php Show that $3p^2=q^2$ implies $3|p$ and $3|q$ I want to know how to get size of objects like a string, integer, etc. in Python.  Related question:   I am using an XML file which contains size fields that specify the size of value. I must parse this XML and do my coding. When I want to change the value of a particular field, I will check the size field of that value. Here I want to compare whether the new value that I'm gong to enter is of the same size as in XML. I need to check the size of new value. In case of a string I can say its the length. But in case of int, float, etc. I am confused. Batch export Photoshop layers to individual PNG files What's the proof that the Euler totient function is multiplicative? That is,     why is $\varphi(AB)=\varphi(A)\varphi(B)$, if $A$ and $B$ are two coprime positive integers?   It's not just a technical trouble—I can't see why this should be, intuitively: I bellyfeel that its multiplicativity should be an approximation at most. And why the minimum value for $\varphi (n)$ should be $ \frac{4n}{15} $ completely passes me by. "He added" vs. "Did he add" (inversion after negative adverb) What could happen if I use 2 cycle fuel in a 4 cycle engine? First off, this is NOT to be confused with the following question:    2 cycle oil is used for my weed eater. There is no way I could use a whole gallon in a year.  Sometimes I run out of gasoline for my 4 cycle lawn mower, and that 2 cycle mix fuel is sitting there getting really old.  If I run that in the lawn mower, am I simply going to create extra pollution or could the 2 cycle fuel cause any damage to the 4 cycle engine? Round double to 3 points decimal When is JavaScript's eval() not evil? Find $x$ in   $$ \Large 2 = x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}}$$   A trick to solve this is to see that $$\large 2 = x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}} \quad\implies\quad 2 = x^{\Big(x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}}\Big)} = x^2 \quad\implies\quad x = \pm \sqrt{2} $$  Are these solutions correct? If not, why? If yes, are there other solutions?    PS: An extension of this discussion can be found in In Harry Potter, what kind of home life do those born to a wizard and a Muggle have? Should the weight of downvotes be increased? Flirtatious Primes Here's a possibly interesting prime puzzle. Call a prime $p$ flirtatious if the sum of its digits is also prime. Are there finitely many flirtatious primes, or infinitely many? After removing any part the rest can be split evenly. Consequences? What's usually the effective visual resolution of typical Hollywood Movies? I'm really surprised by the quality of the visuals in some new movies this year (2018). I'm curious to know in what resolution these cameras typically record to experience this quality on a cinema screen?