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# An absolutely convergent series

Let be a sequence of real numbers such that for all real sequences with the series converges. Prove that the series also converges.

Solution

Suppose on the contrary that the series diverges. Define the sequence of the naturals as follows. We set . Having defined  we define to be the least natural that is greater than and satisfies . The sequence is well defined because the series diverges.

For we define . Then we note that and