If $f(x_{n_k})\to L$ and $(x_n) \to b$ does $f(x_n)\to L$?

Question

I need something like to finish a bigger proof concerning a left hand limit. $f$ is bounded and nondecreasing on $(a,b)$.

the related question is here Prove that if $f$ is bounded and nondecreasing on $(a,b)$ then lim $f(x) $as $x$ approaches $b$ from the left exists. which I provide for context and in case this question is the wrong approach to solving that one. I can't see how to use the hints given to me there because of this issue I raise here.

Answer 1

You want to show that $f(x_n)\to L$. Let $\epsilon>0$. Then there is an integer $k$ such that

$$|L-f(x_{n_k})|< \epsilon$$

as $f(x_{n_k}) \to L$. As $x_{n_k} < b$ and $f$ is nondecreasing, we have

$$L-\epsilon <f(x_{n_k}) \leq L $$

Now fix this $k$. Then as $x_n \to b$, there is $N$ such that

$$ x_{n_k} <x_n < b$$

for all $n\geq N$. Now what can you say about $f(x_n)$ for $n\geq N$? (You need to use that $f$ is nondecreasing).