Prove that $\displaystyle\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\cdots\left(1-\frac{1}{9\,999^2}\right)\left(1-\frac{1}{10\,000^2}\right)=0.500\,05$    Here are all my attempts to solve this problem:  So the first thing I thought about is to transform the expression of the form $$\left(1-\frac{1}{n^2}\right)$$  To the expression: $$\left(\frac{n^2-1}{n^2}\right)$$  But in evaluating the product $$\prod_{k=2}^n    \frac{n^2-1}{n^2}$$ seems way complicated than what I'm capable of, I don't know how to evaluate products even if I know that notation.  The other idea is to transform again $$\left(1-\frac{1}{n^2}\right)\to \left(1-\frac{1}{n}\right)\left(1+\frac{1}{n}\right)$$  But this isn't helpful in any way.  My other attempts were to try to see what happens when I start evaluating this sum, and it turns out that a lot of things cancel out but again no result.  I'll be happy if someone could guide me to solve this problem. (I feel that there is some kind of symmetry that I should remark, a symmetry that would allow me to cancel things out and to have my final)
Evaluate the infinite product $$\lim_{ n\rightarrow\infty }\prod_{k=2}^{n}\left ( 1-\frac{1}{k^2} \right ).$$  I can't see anything in this limit , so help me please.
IS or ARE? "The only thing that I want you to hit right now IS/ARE the books" Why are integers subset of reals? In most programming languages, integer and real (or float, rational, whatever) types are usually disjoint; 2 is not the same as 2.0 (although most languages do an automatic conversion when necessary). In addition to technical reasons, this separation makes sense -- you use them for quite different purposes.  Why did they choose to say $\mathbb{Z} \subset \mathbb{R}$ in math? In other words, why are 2 and 2.0 considered the same?  When you are working in $\mathbb{R}$, does it make any difference whether some elements, eg. 2.0, also belong to $\mathbb{Z}$ or not? "Reality" of length contraction in SR Why does $a^n - b^n$ never divide $a^n + b^n$? How to decide if an effect is magical for Magic Resistance? Some creatures have Magic Resistance:     The [...] has advantage on saving throws against spells and other magical effects.   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Proof involving absolute value and maximums Given the definition for any real numbers a and b, the max function is   $\max\{a, b\} =  \begin{cases}                                    a \text{ if } a \geq b \\                                    b \text{ if } a &lt; b \\   \end{cases} $  Lemma: for any two real numbers $a\text{ and }b, a \leq \max\{a,b\}$ and $ b \leq \max\{a,b\}$.  Prove that for real numbers, $a, b$, and $x$, if a $\leq x \leq$ b then $|x| \leq \max\{|a|,|b|\}$.  So far, I have  $x \geq 0$ or $x &lt; 0$ by the law of trichotomy  Case 1: $x\geq 0$  $|x| = x$ by the definition of absolute value  and I'm not sure where to go from there. Any hints? For $0 < x < \pi/2 $, prove that $x + \frac{x^{3}}{3} <\tan x $ I proved as follows:  Let $f(x) = \tan x -  x - \frac{x^{3}}{3}.$ Then $f '(x) = \sec^{2}x - 1 - x^{2}$ and $f''(x) = 2\sec^{2}x\tan x-x &gt; \tan x - x,$ since $\\tan x &gt; x,$ $f''(x) &gt; 0$ so $f'(x)$ is increasing. Since $f '(0) = 0,$ $f '(x) &gt; 0$ for all $x \in (0, \frac{\pi}{2}).$ So $f(x)$ is increasing. since $f(0) = 0,$ $f(x) &gt; 0$ for all $x \in (0, \frac{\pi}{2}).$  But my proof seems complex. Is there any simpler way to prove this? Due to the implementation of Java generics, you can't have code like this:   public class GenSet&lt;E&gt; {     private E a[];      public GenSet() {         a = new E[INITIAL_ARRAY_LENGTH]; // error: generic array creation     } }   How can I implement this while maintaining type safety?  I saw a solution on the Java forums that goes like this:  import java.lang.reflect.Array;  class Stack&lt;T&gt; {     public Stack(Class&lt;T&gt; clazz, int capacity) {         array = (T[])Array.newInstance(clazz, capacity);     }      private final T[] array; }   But I really don't get what's going on. Is there a way to buy SO reputation? How do I set a background image while I am editing? How to learn physics effectively and efficiently VMware won't work after Ubuntu Upgrade I upgraded to Ubuntu 15.10 and now if I want to open VMware Player it doesn't do anything. I also uninstalled and installed it again, still nothing.  What can I do?