Convergent Series

analysis

Boundedness

All Convergent Series are limited.

Let $(a_n{)}_{n\ge 0}$ be a sequence with ${\lim }_{n\to \infty }{a}_n=a$ and $\epsilon >0$. Then almost all $a_n$ lie in $U_{\epsilon }(a)$ (epsilon neighbourhood of $a$). Let the terms with $a_n\notin U_e(a)$ be $a_{n_1},a_{n_2},\dots ,a_{n_k}$. Let $\bar{\epsilon }>{\max }_{i=1,\dots ,k}|a-a_{n_i}|$ (or $\bar{\epsilon }=\epsilon$ in the case $k=0$). Then in particular $\epsilon \ge \epsilon$, and therefore all sequence elements lie in $U_{\epsilon }(a)$. The interval bounds $a-\bar{\epsilon }$ or $a+\bar{\epsilon }$ then lower and upper bounds, respectively, of $(a_n{)}_{n\ge 0}$.