Q:

In the proof of Bolzano Weierstrass thm.  about converging subsequences. 

Just one line is giving me trouble:
$$(2^{-l+1} + 2^{-l} + 2^{-l-1}+ \cdots + 2^{-m+2} )\cdot M$$
$$=[(2^{-l+2} - 2^{-l+1}) + (2^{-l-1} - 2^{-l}) + \cdots +(2^{-m+3} - 2^{-m+2}) ]\cdot M$$
Hows that? 

A:

Its from the fact that $$2^{k+1} - 2^k = 2^k(2-1) = 2^k$$
Therefore each power of $2$ is expanded as a difference. 
i.e. 
$$2^{-l+1} = 2^{-l+2} - 2^{-l+1}$$
and so on....

