We consider decreasing sequence consist of nonnegative numbers

and corresponding series


We should show that this series satisfies Cauchy criterion and, therefore, convergent:

I found following proof in Web:

If

:

Therefore:

provided

is large.
I don't like this expression:

I know that

, where

- positive constant, but it's not the reason for above inequality because we haven't proved that

can't attain very big values.
For example, if

then

I don't know how to prove that

Could You give me hint, please?