Question on proof of unit ball in $C([0, 1])$ not being compact

Question

Take the sequence $f_n(t)=t^n$, $0\le t\le 1$. Then $\{f_n\} \subset \overline{B(0,1)}$, but we have no subsequence of $\{f_n\}$ converging in $C([0,1])$. So the unit ball is not compact in $C([0,1])$?

Could someone elaborate on this? Why do we have no subsequence of $\{f_n\}$ converging in $C([0,1])$? What does such a subsequence look like? Where does the subsequence converge if it does not converge in $C([0,1])$?

Answer 1

• Note $$f(x) = \lim_{x \to \infty} f_{n}(x) = \left\{\begin{array}{cc} 0, & 0\leq x < 1 \\ 1, & x =1 \end{array}\right.$$ Can you see now why $f \not \in C[0,1]$