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)

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

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

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user061703 · Accepted Answer · 2018-05-18T12:30:04.813

0

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 $n\in\mathbb{Z^+}$)

edited May 18 '18 at 12:30

answered May 18 '18 at 12:28

user061703

Why $1$ becomes $2^n$? – user560980 May 18 '18 at 12:29
This is an assumption. It does not prove anything. It is like saying $2 < 1$ so $2+1<1+1$ proved :/ Edit: Ok after your edit, I take that back :) – Mr Pie May 18 '18 at 12:30

1 Answers1