- #1
-Dragoon-
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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?