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We prove by induction on `n`

forallninN:n< 2^{n}.

The *induction base* holds because 0 < 1 = 2^{0}.

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

(1)We have to show (n< 2^{n}.

(2)By (1) we haven+1 < 2^{n+1}.

(3)and thereforen+1 < 2^{n}+1

(4)which implies (2).n+1 < 2^{n}+1 <= 2^{n}+2^{n}= 2*2^{n}= 2^{n+1}

Author: Wolfgang Schreiner

Last Modification: November 24, 1999