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 4 of 4: Complete the induction
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b1+⋯+bn+an+1n+1≥an+1b_1+\cdots+b_n+\frac{a_{n+1}}{n+1}\ge a_{n+1}
Detailed analysis

Divide the preceding bound by n+1 and add a(n+1)/(n+1). This gives b1+...+bn+ a(n+1)/(n+1) >= a(n+1), exactly the assertion for n+1. Induction proves the result for every positive n.