Let $f_{1}(x),f_{2}(x),\dots,f_{n}(x)$ be $n$ bounded and uniformly continuous functions on $\mathbb{R}$. Prove that their product $f_1(x)f_2(x)\cdots f_n(x)$ is also a uniformly continuous function on $\mathbb{R}$  My approach is to use mathematical induction. At the last step, I need to prove that the product of two uniformly continuos functions (which are the product of the first $n-1$ uniformly continuous functions by inductive hypothesis and the $n^{\text{th}}$ function), which doesn't require me to use the fact the functions are bounded.   So, I sense that this approach is not correct (but I'm not sure why) as I didn't use the fact that the functions are bounded. If it's really wrong, how do I make use of the fact that the functions are bounded? Thank you :)
prove that if two real valued  functions are uniformly continuous on some interval $I$  and each is bounded on $I$ then their product $f\cdot g=f(x)\cdot g(x)$ is also uniformly continuous on $I$ . Is boundedness of each function  on $I$ is  necessary for the product
Are PDO prepared statements sufficient to prevent SQL injection? I'd like to be able to tail the output of a server log file that has messages like:  INFO SEVERE   etc, and if it's SEVERE, show the line in red; if it's INFO, in green.  What kind of alias can I setup for a tail command that would help me do this? How can I get a DisplayLink USB monitor work under Ubuntu? I have a DisplayLink-based external USB monitor, which has both power and data over USB, and seems to work perfectly under Windows 7, but only can display a text console under Ubuntu 10.10, and that I can only use when I am actually switched to it. So the only Use I can have so far is to have some text-based monitoring or console that I can watch while working on the graphic display.   I know there are some development done for DisplayLink, but I never could get it to actually run properly.   Yes, I tried the detailed setup described in other , but they did not work, and instead crashed my X that I had to restore.   Merci :-) How to show that all points are inside of unit circle? There are $n$ points on the plane. Any $3$ of them are inside of a unit circle. How to show that all points are inside of unit circle?  It is needed to prove that if there is a unit circle for each $3$  points which contains them, there is also a unit circle which contains all $n$ points. How can I make an object follow a path exactly I want to make this sphere follow a path, but everything I try creates a giant distance from the actual path. I guess what I'm asking is how do I get the blue line extending from the path to the sphere to be basically 0 at all times? Error entering equation in DSolve How can I apply a -p0 patch from any working directory? Who built the fortress of Dol Goldur, and when was it built? And was it always a place for evil, or Sauron just occupied it?  Despite the fact that Mordor is much better fortified; artificially by the old fortifications,  and geographically, as it is surrounded by the mountains. Also considering the presence of mount doom that seem to enhance Sauron's power, Sauron chose to start his campaign from Dol Goldur, despite Mordor being ready for quite some time.  Why did he choose this enigmatic fortress, and not go straight for Mordor or even go earlier? In iPhone, you can easily change the default ringtone (Settings → Sounds → Ringtone) and set a custom ringtone for individual people (Contacts → [some person] → Edit → Ringtone).   But can you set a custom tone for a group of contacts? (E.g. "family" or "workmates".)  I think that would be handy—and quite basic functionality, so to speak—but I couldn't find a way to do this through the iPhone UI. Is it possible at all (without jailbreaking)? Infinite product with sign changes. For infinite products I know the following theorems:  Theorem I. If $a_n \geq 0$ for all $n$ then $\prod (1+a_n)$ converges if and only if $\sum a_n$ converges.  Theorem II. If $-1 &lt; a_n \leq 0$ for all $n$ then $\prod (1+a_n)$ converges if and only if $\sum a_n$ converges.  My question is what are necessary and sufficient conditions (possibly including more restrictions than in I and II) for $\prod (1+a_n)$ convergence/divergence as related to  $\sum a_n$ convergence/divergence  if the sequence of terms $a_n$ can switch signs repeatedly but always stay greater than $-1$.  I already know the trivial equivalence of $\prod(1+a_n)$ and $\sum\ln(1+a_n)$. new array.sty and colortbl.sty incompatibility? How to find which RAM to get for my computer? I don't have any knowledge in hardware and would like to know which RAM should I buy for my laptop.  How do I discover what RAM I need &amp; how much my machine can take?  I have a Razer Blade 15 (RZ09-02705F76) 2018 but don't know if there is a specific type or format of RAM that suits it and don't know where to find this information. Is there a word or phrase for something that one might wish exists, but most certainly doesn't? Given a set of vectors $v_1 \cdots v_n$, the $n\times n$ Gram matrix $G$ is defined as $G_{i,j}=v_i \cdot v_j$  Due to symmetry in the dot product, $G$ is Hermitian.  I'm trying to remember why $|G|=0$ iff the set of vectors are not linearly independent. Explode string into array with no empty elements? PHP's explode function returns an array of strings split on some provided substring. It will return empty strings when there are leading, trailing, or consecutive delimiters, like this: var_dump(explode('/', '1/2//3/')); array(5) {   [0]=&gt;   string(1) &quot;1&quot;   [1]=&gt;   string(1) &quot;2&quot;   [2]=&gt;   string(0) &quot;&quot;   [3]=&gt;   string(1) &quot;3&quot;   [4]=&gt;   string(0) &quot;&quot; }  Is there some different function or option or anything that would return everything except the empty strings? var_dump(different_explode('/', '1/2//3/')); array(3) {   [0]=&gt;   string(1) &quot;1&quot;   [1]=&gt;   string(1) &quot;2&quot;   [2]=&gt;   string(1) &quot;3&quot; } User’s Guide vs Users’ Guide I’ve been looking over what has been posted regarding the use of ’s.   I used to be a Technical Writer (years ago). The title of one of our training documents was Users’ Guide.  Once, a coworker said every time he saw that title he expected users to start coming.    Never made any sense to me, but I have to admit that the majority of us didn't understand the use of s’ in place of ’s.  What is the difference? Find out if any files were exported from my MacBook NSInvalidUnarchiveOperationException: Could not instantiate class named NSLayoutConstraint Is it possible to set DNS for mobile network Why there is a $\frac{1}{2}$ in the distance formula $d=\frac{1}{2}at^2$? I'm preparing for my exam, but I have difficulties in perceiving why there is a $\frac{1}{2}$ in the distance formula $d=\frac{1}{2}at^2$?