Say $x_n$ is a bounded sequence in a closed interval [a, b] and $y_n$ is the convergent subsequence. how can I prove that the limit of the convergent subsequence must be in the interval [a, b]?
Since $x_n$ is bounded within the interval [a, b]. Thus, for every subsequence $y_n$ of $x_n$, $y_n$ must be in the closed interval as well. Thus, the limit of convergent subsequence must be in the interval. Is my proof correct?
Best Answer
Somehow you need to use the closeness of $[a,b]$
You may approach by contradiction. If the limit of the convergent subsequence is not in $[a, b]$, then either the limit is larger than $b$ or less than $a$
Now both cases leads to contradiction because you may choose a small neighborhood of the limit which does not intersect $[a,b]$ contrary to the terms being in $[a,b]$