Question

Let $(s_{n})$ be a sequence that converges. [(a)] Show that if $s_{n} \ge a$ for all but finitely many $n$, then $\lim s_{n} \ge a$. [(b)] Show that if $s_{n} \le b$ for all but finitely many $n$, then $\lim s_{n} \le b$. [(c)] Conclude that if all but finitely many $s_{n}$ belong to $[a, b]$, then $\lim s_{n}$ belongs to $[a,b]$.

Name: let sn be a sequence that converges a show that if sn ge a for all but finitely many n then lim sn ge a b show that if sn le b for all but finitely many n then lim sn le b c conclude that if all
Uploaded: 2021-04-30T09:33:47-08:00
Duration: 8 min 40 s
Channel: Aman Gupta
Description: let sn be a sequence that converges a show that if sn ge a for all but finitely many n then lim sn ge a b show that if sn le b for all but finitely many n then lim sn le b c conclude that if all

          Let $(s_{n})$ be a sequence that converges.
         [(a)] Show that if $s_{n} \ge a$ for all but finitely many $n$, then $\lim s_{n} \ge a$.
[(b)] Show that if $s_{n} \le  b$ for all but finitely many $n$, then $\lim s_{n} \le b$.

[(c)] Conclude that if all but finitely many $s_{n}$ belong to $[a, b]$, then
                $\lim s_{n}$ belongs to $[a,b]$.

Added by Michael L.

Calculus for AP

Jon Rogawski & Colin Adams 3rd Edition

Chapter 10

Instant Answer

Solved by Expert Aman Gupta

04/30/2021

Step 1

Step 1: Given that $(s_{n})$ is a sequence that converges, we want to show that if $s_{n} \ge a$ for all but finitely many $n$, then $\lim s_{n} \ge a. Show more…

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Let $(s_{n})$ be a sequence that converges. [(a)] Show that if $s_{n} \ge a$ for all but finitely many $n$, then $\lim s_{n} \ge a$. [(b)] Show that if $s_{n} \le b$ for all but finitely many $n$, then $\lim s_{n} \le b$. [(c)] Conclude that if all but finitely many $s_{n}$ belong to $[a, b]$, then $\lim s_{n}$ belongs to $[a,b]$.