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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.We have to show (induction step)
(2) n+1 < 2n+1.By (1) we have
(3) n+1 < 2n+1and therefore
(4) n+1 < 2n+1 <= 2n+2n = 2*2n = 2n+1which implies (2).