Cauchy convergence criteria

Question

I was reading about convergence of a series and there is a part regarding Cauchy convergence criteria and there is an equivalence I don't understand. What I don't understand is why for $n\ge m \ge N$ this:$$\sum_{k=m}^n a_k \lt \epsilon $$ is equivalent to this: $$\lim_{n\to\infty,m\to\infty} \sum_{k=n}^{n+m}a_k = 0$$