The question and its answer is given below:
But I am not understanding the sequence of steps in the solution what are the ideas he using, for example:
1- In the second line why he is using a subsequence ${f_{n_{k}}}$ and he did not use only apoint.
2-In the forth line why we are sure that there exists such $x_{k}$ for each $k$, and why we are doing this?
3-Also I see we arrived at a contradiction to that $f_{n}$ converges pointwisely to $f$, is my understanding correct?
Could anyone clarify these discrepancies for me please?