The Key thing to remember is that a series is a sequence of partial sums
The Cauchy Crterion for a series states that for EVERY epsilon > 0, there exists an N a natural number such than for n,m > N
but what is this exactly? Well
These are both members of the sequence of partial sums
Now since we are talking about a sequence(of partial sums) we can use the Cauchy Criterion for sequences.
I hope this clears it up.