## IMC 2016 – Day 2 – Problem 6

**Problem 6.** Let be a sequence of positive real numbers satisfying . Prove that

*Sketch of proof:* We can write the sum to be bounded as

Rearranging terms (we can do it since the series is absolutely convergent) we note that we have

Taking a look at the hypothesis we note that it is enough to prove that

Since this doesn’t work by induction right away (try it…) let’s try and prove something stronger:

Since the induction will work.

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Categories: Analysis, Inequalities, Olympiad, Problem Solving
Analysis, IMC, Inequalities, series

I did the very same thing with difference in the fact that i used able’s summation by parts formula at the start . Eventually i was lead to prove something very similar to last of inequalities of this post.