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Absolute convergence implies Cauchy

An absolutely convergent series has Cauchy partial sums.

Absolute convergence implies Cauchy: Let ∑n=1∞an\sum_{n=1}^\infty a_n be a series of real or , and let sn=∑k=1naks_n=\sum_{k=1}^n a_k be its . If

∑n=1∞∣an∣\sum_{n=1}^\infty |a_n|

converges, then (sn)(s_n) is a ; equivalently, for every ε>0\varepsilon>0 there exists NN such that for all integers m>n≥Nm>n\ge N,

∣∑k=n+1mak∣<ε.\left|\sum_{k=n+1}^m a_k\right|<\varepsilon.
Remarks

This is the series form of the and is a standard route to .