In Kreyszig's proof (Functional analysis) that $l^p$ is complete, one defines a Cauchy sequence such that for every $\epsilon>0$, there is $m,n>N$ s.t. $(\sum_{j=1}^\infty|\xi_j^{(m)}-\xi_j^{(n)}|)^{1/p}$ ($x_m=(\xi_1^{(m)},\xi_2^{(m)},\dots)$). This implies $|\xi_j^{(m)}-\xi_j^{(n)}|<\epsilon$ for all $j$. From here one can conclude $\xi_j^{(m)}\rightarrow \xi_j$ as $m\rightarrow\infty$. Now we have $$ \sum_{j=1}^k |\xi_j^{(m)} - \xi_j^{(n)}|^p<\epsilon^p\;(1) $$ and by letting $n\rightarrow \infty$ one obtains for $m>N$ $$ \sum_{j=1}^k |\xi_j^{(m)} - \xi_j|^p\leq\epsilon^p.\;(2) $$
Besides the obvious ($a<b$ implies $a\leq b$), what is the insistence of the author to stress that the limit $n\rightarrow\infty$ converts the "<" in (1) to "$\leq$" in (2)? More specifically, what is the proof to go from (1) to (2)?