Prove that $n<2^n$ for every positive integer $n$

Question

Prove that $n<2^n$ for every positive integer $n$.

$P(1): 1 < 2^1$

$n+1<2^n+1$ for induction hypothesis

$n+1<2^n+2^n$ I can't undertstand this passage (this is from my teacher's slide)

$n+1<(2)2^n$

$n+1<2^{n+1}$

Answer 1

Assume that $n<2^n$ is true.

We need to prove that $n+1<2^{n+1}$ is also true.

This is true because $n<2^n\Rightarrow n+1<2^n+1<2^n+2^n<2^n\times2=2^{n+1}$

($2^n>1$ for all $n\in\mathbb{Z^+}$)