Also why do we assume m>n?

Assuming that \(\displaystyle m>n\) can be done without the lost of any generality(i.e. it does no harm).

Many of us simply I no idea what you know about sequences & series. Therefore, makes giving you help very difficult.

Here are sone facts: if the series \(\displaystyle \sum\limits_{k = 1}^\infty {{S_k}} \) converges and \(\displaystyle c>0\) then there exists a positive integer \(\displaystyle N\) such \(\displaystyle \left| {\sum\limits_{k = N}^\infty {{S_k}} } \right| < c\).

That tells us that the "tail-end" of a convergent series is small. So any part of the tail is very small.

That is the driving concept behind a Cauchy sequence.

Now what we have is \(\displaystyle \left| {\sum\limits_{k = N}^\infty {{2^{-k}}} } \right| \) is convergent.

Can turn that into a Cauchy sequence ?