<s>[INST] Show by induction: $(1+\frac{1}{n})^{n}Show by induction that for all natural numbers n>3
$(1+\frac{1}{n})^n<n$
Let $(1+\frac{1}{n})^n<n$ be true ! 
We show that $(1+\frac{1}{n+1})^{n+1})<n+1$
$(1+\frac{1}{n})^n(1+\frac{1}{n})<n(1+\frac{1}{n})$ 
$(1+\frac{1}{n})^{n+1}<n+1$
So now we just need to show that 
$(1+\frac{1}{n+1})^{n+1}<(1+\frac{1}{n})^{n+1}$
So here , If I show that : 
$(1+\frac{1}{n+1})<(1+\frac{1}{n})$ 
Does that mean im done with exercise ?
 [/INST] You have to prove for $n=4$ in order that mathematical induction work! To show the desired implication, note that if $n \geq 4$ is such that 
$$
\bigg( 1 + \frac{1}{n} \bigg)^{n} < n,
$$
then
$$
\bigg( 1 + \frac{1}{n+1} \bigg)^{n+1} = \bigg( 1 + \frac{1}{n+1} \bigg)^{n} \bigg( 1 + \frac{1}{n+1} \bigg) < \bigg( 1 + \frac{1}{n} \bigg)^{n}\bigg(1 + \frac{1}{n+1} \bigg) < n + \frac{n}{n+1} < n+1.
$$
Note also that this implication makes no use of the hypothesis that $n \geq 4$, so if you do not check the initial proposition then a wrong conclusion could be drawn!
</s>