- #1

-Dragoon-

- 309

- 7

## Homework Statement

Prove and show that 2n ≤ 2^n holds for all positive integers n.

## Homework Equations

n = 1

n = k

n = k + 1

## The Attempt at a Solution

First the basis step (n = 1):

2 (1) ≤ 2^(1) => 2 = 2.

Ergo, 1 ϵ S.

Now to see if k ϵ S:

2 (k) ≤ 2^k

But, k ϵ S implies k + 1 ϵ S:

2(k + 1) ≤ 2^(k + 1)

2k + 2 ≤ 2^k · 2^1

Now, where do I go from here to prove this formally and that k + 1 ϵ S, thus proving that 2n ≤ 2^n holds for all positive integers n?