A question with the sequence $e_{n}=\left(1+\frac{1}{n}\right)^{n}$ Prove that 
$a$) the following sequence is increasing
$$e_{n}=\left(1+\frac{1}{n}\right)^{n},\quad n\ge1;$$
$b$) the inequality below holds
$$e_{n} \leq3,\quad n\ge1.$$
 A: For the first part use the binomial theorem and show that each component is non-decreasing and some are increasing.
For the second part you can use the same binomial expansion term by term to show that $$\left(1+\frac1n\right)^n<1+1+\frac12+\dots\frac 1{r!}+\dots<1+1+\frac 12+\dots\frac 1 {2^r}+\dots$$ and sum the geometric progression.
A: In order to prove that the given sequence is strictly increasing, we are to demonstrate $e_{n+1} > e_n$:
\[ 
\bigg(1+ \dfrac{1}{n+1}\bigg)^{n+1} > \bigg(1 + \dfrac{1}{n} \bigg)^n.
\]
Let's rewrite the inequality above as:
\[ \bigg( \dfrac{1 + \dfrac{1}{n+1}}{ 1 + \dfrac{1}{n}} \bigg)^n > \dfrac{1}{1 + \dfrac{1}{n+1}}.
\]
The right-hand side equals
\[ \dfrac{1}{1 + \dfrac{1}{n+1}} = \dfrac{n+1}{n+2} = 1 - \dfrac{1}{n+2}.
\]
Now, let's focus on the left-hand side:
\[ \bigg( \dfrac{1 + \dfrac{1}{n+1}}{ 1 + \dfrac{1}{n}} \bigg)^n = \bigg( \dfrac{(n+2)/(n+1)}{(n+1)/n} \bigg)^n = \bigg( \dfrac{n(n+2)}{(n+1)^2} \bigg)^n = \bigg( 1 - \dfrac{1}{(n+1)^2}\bigg)^n.
\]
By the Bernoulli's inequality, the following holds:
\[ \bigg( 1 - \dfrac{1}{(n+1)^2}\bigg)^n \geq 1 - \dfrac{n}{(n+1)^2}
\]
Now it's purely technical to show the desired inequality
\[ 1 - \dfrac{n}{(n+1)^2} > 1 - \dfrac{1}{n+2},
\]
because
\[ \dfrac{n}{(n+1)^2} < \dfrac{1}{n+2}.
\]
