Suppose $(S_n)$ is an infinite sequence of real numbers and that $a, b \in \mathbb{R}$ are such that $S_n \in [a, b]$ for all but finitely many $n$. Prove that if $(S_n)$ converges then $\lim_{n\to\infty} S_n \in [a, b]$. Do not use the limit laws.
So I've started as follows: Let $N= \max \{ n \;|\; S_n \not\in [a,b] \}$. This exists and is finite because $S_n$ is outside $[a,b]$ for finitely many $n$. Now suppose for a contradiction that $\lim_{n\to\infty} S_n = S$ and $S$ is not in $[a,b]$.
At this point I need to pick some epsilon that will give me a contradiction using the definition of a limit, but I am completely at a loss on how to do that. Any help would be appreciated!
$[a, b]$ is an closed interval, so every convergent sequence in this interval converges to the point in this interval.
And, Every subsequence of an convergent sequence is converges to the same point.
Now, dropping out the elements from {S_n} which are lying outside the interval $[a, b]$. This is one of the possible subsequence having all of it's elements lying inside the closed interval $[a, b]$. So it is converges to the point in this interval.