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Math 225 004 Indirect Proofs

1. Write a proof by contraposition to show that for any real number x, if x2 + 2x < 0,then x

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Proof. Letnbe a natural number. Assume that n

n+ 1

n

n+ 2. Son(n+2) n(n+1).

This is equivalent to n + 2 n + 1 sincen is positive. Subtracting n from both sides ofthe inequality results in 2 1, which is a contradiction. Thus ifn is a natural number,

then n

n+ 1 >

n

n+ 2.