Q:

Does convergence of $f(x_n)$ imply convergence of $f(x_{n_r})$

Let $I=[a,b]$. Suppose $f:I\to R$ is a continuous function on $I$ and we are given that there is a bounded sequence $(x_n)\subseteq I$ such that $(f(x_n))$ converges to $L$.
By BWT, we know that there exists a convergent subsequence $(x_{n_r})$ converging to some point, say $c\in I$.  
Can we conclude from here, that $f(c)=L$ ?
I think this is true, but not able to give rigorous argument.  

A:

If $f(x_n)$ converges to $L$ then the subsequence $f(x_{n_r})$ also converges to $L$. By continuity $f(x_{n_r})$ also converges to $f(c)$. Hence $L=f(c)$.

