<s>[INST] Prove that $a_n = \{\left(1+\frac{1}{n}\right)^n\}$ is bounded sequence, $ n\in\mathbb{N}$ How to prove the following:
$a_n = \left\{\left(1+\frac{1}{n}\right)^n\right\}$ is bounded sequence, $ n\in\mathbb{N}$
 [/INST] Let me denote $S_n=\{(1+\frac{1}{n})^n\}$
\begin{align*}
\bigg(1+\frac{1}{n}\bigg)^n &=1+ {_nC_1} \frac{1}{n}+ {_nC_2} \bigg(\frac{1}
{n}\bigg)^2+..........+{_nC_n} \bigg(\frac{1}{n}\bigg)^n\\
&=1+ n \frac{1}{n}+\frac{n(n-1)}{2!} \bigg(\frac{1}{n}\bigg)^2+ \frac{n(n-1)(n-2)}{2!} \bigg(\frac{1}{n}\bigg)^3+..........+\frac{n(n-1)(n-2)....(n-(n-1))}{n!}\bigg(\frac{1}{n}\bigg)^n\\
&=1+ 1+ \frac{(1-\frac{1}{n})}{2!} + \frac{(1-\frac{1}{n})(1-\frac{2}{n})} {3!}+..........+\frac{(1-\frac{1}{n})(1-\frac{2}{n}).....(n-\frac{n-1}{n})} {n!}\\
&<1+ 1+ \frac{1}{2!} + \frac{1}{3!}+..........+\frac{1}{n!}\\
&<1+ 1+ \frac{1}{2} + \frac{1} {2^2}+..........+\frac{1}{2^{n-1}}\\
&=1+\frac{1-(\frac{1}{2})^n}{1-\frac{1}{2}}\\
&=1+2\bigg(1-\frac{1}{2^n}\bigg)\\
&=3-\frac{1}{2^{n-1}}\\
&<3\\
\end{align*}
$S_n$ is bounded by 3.
Also note that $S_n>2$. 
Here I'm not writing the reason in every step i hope u will understand.
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