User:Genia4~enwikibooks

Proof:

Let $$\epsilon>0$$

According to the Cauchy criterion for series convergence, exists $$N$$ so that for all $$N<m,n$$:

$$\sum_{k=n}^{m}|a_{k}|<\epsilon$$

We know that:

$$|\sum_{k=n}^{m}a_{k}|\leq\sum_{k=n}^{m}|a_{k}|$$

And then we get:

$$|\sum_{k=n}^{m}a_{k}|\leq\sum_{k=n}^{m}|a_{k}|<\epsilon$$

Now we get:

$$|\sum_{k=n}^{m}a_{k}|<\epsilon$$

Which is exactly the Cauchy criterion for series convergence.

$$Q.E.D$$