Convergence of an alternating series

• I
Consider a sequence with the ##n^{th}## term ##u_n##. Let ##S_{2m}## be the sum of the ##2m## terms starting from ##u_N## for some ##N\geq1##.

If ##\lim_{N\rightarrow\infty}S_{2m}=0## for all ##m##, then the series converges. Why?

This is not explained in the following proof:

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