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

thus the series diverges. This leads to contradiction hence the result.