I proved the first inequality $0<e-(1+\frac{1}{n})^n$ but now struggle with second part. I tried mathematical induction:

1. For $n=1$ $e-2<\frac{3}{1}$
2. $e-(1+\frac{1}{n})^n<\frac{3}{n} \Rightarrow e-(1+\frac{1}{n+1})^{n+1}<\frac{3}{n+1}$