# How can I proof that f(x) is not pointwise convergence? And How can I draw this function? [closed]

How can I prove that $f(x)$ is not pointwise convergent? And how can I draw this function?

f_n(x) = \left\{\begin{aligned} &nx &&: 0 \ge x \ge \frac{1}{n}\\ &2-nx &&: \frac{1}{n} \ge x \ge \frac{2}{n}\\ &0 &&: \frac{2}{n} \ge x \ge 1 \end{aligned} \right.

• Are you sure you copied the third line of the definition of $f_n$ correctly? Did you mean to ask about pointwise convergence of $f_n$ rather than of $f(x)f(x)$? Nov 30, 2016 at 21:17
• You right Blass, that was mistake. It should be f_n(x) = 0 Nov 30, 2016 at 21:20
• $f_n(\frac 1n) = 1$ for all $n.$ The functions starts at 0, rises to 1, falls back to 0 and is a flat line thereafter. The party hat on the left hand side get steeper and steeper as $n$ becomes large. Nov 30, 2016 at 21:33
• You can find picture showing the functions in this question: Convergence and uniform convergence of a sequence of functions. Dec 1, 2016 at 3:16

Hint $$f_n(0)=0$$
for $x>0$ and large enough $n$, $\frac{2}{n}<x \implies f_n(x)=0$.