<s>[INST] $2 < (1 + \frac{1}{n})^n < 3$ for all $n > 2$ Show by induction that $2 < (1 + \frac{1}{n})^n < 3$ for all natural n > 2.


*

*Induction base. For $n = 3$ the inequality is obviously  true.

*Assume that for $n = k$ inequality is true, than i can prove $(1 + \frac{1}{n+1})^{n +1} < 3$.
$$\frac{(n + 1)^{n}}{n^{n}} < 3 \\
1 < \frac{3n^{n}}{(n+1)^{n}} \\
\frac{(n+2)^{n+1}}{(n+1)^{n+1}} < \frac{3 n^{n} (n + 2)^{n+1} }{(n+1)^{2n +1}}, but \\
\frac{3 n^{n} (n + 2)^{n+1} }{(n+1)^{2n +1}} < 3, then \\
\frac{(n+2)^{n+1}}{(n+1)^{n+1}} < 3
$$
How can i prove, that $(1 + \frac{1}{n+1})^{n +1} > 2$
 [/INST] HINT
What about Bernoulli's inequality?
</s>