# Proof by Mathematical Induction

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Prove the following statement:

Prove that if A1+A2+....+An=n then A1A2...An<=1, where A1,A2,...,An are positive real numbers.

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#### Solution Preview

Proof:

When n=1, if A1=n, then A1=1<=1 is true.

Suppose "A1+A2+...+An=n implies A1A2...An<=1" holds for n, then for n+1,

if A1+A2+...+An+A(n+1)=n+1, then we have two cases:

Case 1 (Trivial case): A1=A2=...=An=A(n+1)=1, then

A1A2...AnA(n+1)=1<=1 is ...

#### Solution Summary

A proof is provided by mathematical induction.

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