<s>[INST] Show that $2 < (1+\frac{1}{n})^{n}< 3$ without using log or binommial coefficient $2 <(1+\frac{1}{n})^{n}< 3$
Is it possible to show the inequality without using binommial coefficients thus only by induction? The leftinequality can be shown using bernoulli inequality.
 [/INST] I have found the answer to my problem in a different forum:
We show that $b_n > b_{n+1}$ with $b_n:= (1+\frac{1}{n})^{n+1}$
Proof:
To show: $\frac{b_n}{b_{n+1}}\geq 1$
$\frac{b_n}{b_{n+1}} = \frac{\left(1+\frac{1}{n}\right)^{n+1}}{\left(1+\frac{1}{n+1}\right)^{n+2}} = \frac{1}{1+\frac{1}{n}} \cdot \left(\frac{1+\frac{1}{n}}{1+\frac{1}{n+1}}\right)^{n+2} = \frac{1}{1+\frac{1}{n}} \cdot \left(1+\frac{1}{n(n+2)}\right)^{n+2}$
Exploiting the bernoulli inequality
$\frac{1}{1+\frac{1}{n}} \cdot \left(1+\frac{1}{n(n+2)}\right)^{n+2} \geq 1$
$b_5 < 3$ and $(1+\frac{1}{n})^n < b_n $
Thus $\forall n\geq 5: (1+\frac{1}{n})^n < 3$ 
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