Question

By PMC, prove that inequality $n<2^n$ for all $n\in N$

Answer 1

Step 1: prove for $n=1$

$1<2$

Step 2:

$n+1<2\left(2^n\right)$

$n<2\left(2^n\right)-1$

$n<2^n+2^n-1$

The function $2^n+2^n-1$ is surely higher than $2^n-1$ so if $n<2^n$ is true (induction step), $n<2^n+2^n-1$ has to be true as well.