$\displaystyle(1 + \frac{1}{n})^n < n$ for $n \gneq 3$
yes for $n = 1$ it is true
I assume it is true for $n = k$ and get
$\displaystyle(1+\frac{1}{k})^k < k$
I then go to $\displaystyle(1 +\frac{1}{k+1})^{k+1} < k+1$ and now I spend an hour doodling.