How to show a sequence converges uniformly?

Question

How do you show that a sequence does or does not converge uniformly? Consider:

$f_n(x)$ = {0 if x = 0, n if 0 $\lt$ x $\lt$ $1\over{n}$, 0 if $1\over{n}$$\leq$ x $\leq$ 2

I know that for any given x, {$f_n$}$_{n=1}^\infty$ converges to 0. I also know that it converges to 0 faster or slower depending on the x. For example if x = 1, the sequence has already hit 0 at n=1 and it will stay there until infinity. But if x = $1\over{10}$, it won't hit 0 until n = 10.

Answer 1

If $f_n\rightarrow f$ uniformly, then $\int f_n \rightarrow \int f$. Since the point-wise limit is $0$, the uniform limit should also be $0$.

Now for any $n$ we have,

$$\int_{0}^2 f_n(x)dx = n\times \frac{1}{n}= 1 \neq 0 = \int_0^2f(x)dx$$

Conclusion?

Answer 2

For $n>0,\;\;$ take for example $$x_n=\frac{1}{2n}.$$

we have

$$(\forall n>0)\;\;\; 0<x_n<\frac{1}{n}$$

$\implies$

$$(\forall n>0)\;\;\; f(x_n)=n$$

$\implies$

$$(\forall n>0) \;\;\sup_{x\in[0,2]}|f_n(x)|\geq n$$

$\implies$

$$\lim_{n\to+\infty}\sup_{x\in[0,2]}|f_n(x)-0|=+\infty$$

$\implies$

the sequence of functions $(f_n)$ doesn't converge uniformly to zero at $[0,2]$