Question on pointwise convergence of Sequence of functions

Question

In section 8.2 of textbook "Introduction to real analysis by Bartle" example 8.2.1(c) is as follows:

$f_n : [0,1] →\mathbb{R}$ defined for $n≥2$ by

$$f_n(x) = \begin{cases} n^2x & \text{for $0≤x≤1/n$} \\[6pt] -n^2(x - 2/n) & \text{ for 1/n≤x≤2/n} \\[6pt] 0 & \text{for 2/n≤x<1} \end{cases}$$

then $f_n$ is continuous on $[0,1]$ and hence Riemann integrable______ (upto this i have no problem)

My problem$f_n(x)→0$ for all $x\in\mathbb{R}$ but i unable to see this...

My attempt clearly for $x=0$, $f_n(0)=0$

For $0<x<1/n$, $|f_n(x)-f(x)|=|n^2x|=n^2x$

How, can say this is $<\epsilon$ for sufficiently large $n$?

Further, for other points $x$ how to establish its convergence and how to show integral of $f_n\text{ from $0$ to $1$}=1$ for $n≥2$

Please help and explain this example 8.2.1(c) in detail.

Answer 1

For a fixed $x \in (0,1]$, there exists a positive integer $N > 2/x$.

Thus, for all $n \geqslant N$ we have $x > 2/n$ and $|f_n(x)-0| = f_n(x) = 0$.