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Example

We prove by induction on n

forall n in N: n < 2ⁿ.

The induction base holds because 0 < 1 = 2⁰.

Now we take arbitrary n in N and assume (induction hypothesis)

(1) n < 2ⁿ.

(2) n+1 < 2ⁿ⁺¹.

(3) n+1 < 2ⁿ+1

(4) n+1 < 2ⁿ+1 <= 2ⁿ+2ⁿ = 2*2ⁿ = 2ⁿ⁺¹