MathLabs

Problem 2

Let a1,a2,... be a sequence of real numbers satisfying a(i+j) <= ai+aj for all positive integers i,j. Prove that a1+a2/2+...+an/n >= an for every positive integer n.
Step 2 of 4: Sum the induction hypotheses
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(n+1)b1+(n+1)b2+⋯+(n+1)bn≥2(a1+⋯+an)(n+1)b_1+(n+1)b_2+\cdots+(n+1)b_n\ge2(a_1+\cdots+a_n)
Detailed analysis

Assume b1+...+bk >= ak for every k from 1 through n. Add these n inequalities. Also add the identity b1+2b2+...+n bn=a1+...+an. The coefficient of every bi becomes n+1, giving the displayed inequality.