So through as I've been venturing through baby Rudin I came upon his definition of a cauchy sequence:
A sequence $\{ p_n \}$ in some metric space $X$ is said to be cauchy if $$ \forall \; \epsilon > 0 \; \exists \; N \in \mathbb{N} \; s.t. d(p_n, p_m) < \epsilon \; \forall \; n,m \ge N $$ He then talks about the cauchy criterion for convergence being that a cauchy sequence converges to a point in its contained metric space (similarly this metric space would be called complete). He goes on to say that the cauchy criterion for convergence for series can be restated as the following:
A series $\sum a_n$ converges if and only if $$ \forall \; \epsilon > 0 \; \exists \; N \in \mathbb{N} \; s.t. \left\lvert \sum_{k=n}^{m} a_k \right\rvert \le \epsilon \; \forall \; n,m \ge N $$ Note that these series live in $\mathbb{R}^k$ I can see how one would get
A series $\sum a_n$ converges if and only if $$ \forall \; \epsilon > 0 \; \exists \; N \in \mathbb{N} \; s.t. \left\lvert \sum_{k=n}^{m} a_k \right\rvert < \epsilon \; \forall \; n,m \ge N $$
but I don't understand why the new definition has a $\le$ rather than a $<$
Could anybody advise me as to why?
Thanks in advanced!