Hi,
I've problem with following convergence (in red shape). The second one term is tends to zero it is obvious but why the first one is tends to zero?
Thanks for any help.
You are told that $\displaystyle x_n\to x$ in the sup norm. This means that, given $\displaystyle \varepsilon>0$, there exists N such that $\displaystyle n\geqslant N\ \Rightarrow\ \|x_n-x\|_{\text{sup}} = \sup\{|x_n(t) - x(t)|:t\in X\} <\varepsilon$. Take $\displaystyle t=p_n$ to see that $\displaystyle n\geqslant N\ \Rightarrow\ |x_n(p_n) - x(p_n)|<\varepsilon$. Thus $\displaystyle |x_n(p_n) - x(p_n)|\to0$ as $\displaystyle n\to\infty$.