Prove ${f_n}$ does not converge uniformly to the zero function.

Question

Let ${f_n}$ be a sequence of functions $f_n:(0,1)\to\Bbb R$ defined by $f_n(x)=1/(nx)$. Prove that ${f_n}$ does not converge uniformly to the zero function.

If you could walk me through this to understand it, that would be appreciated.

Answer 1

Pick $\epsilon = 1/2$. Let $n\in\mathbb N$. Then you can always find $x$ close to zero for which $\frac1{nx} > 1/2$.

Answer 2

$f_n\to f$ uniformly on $A\subseteq\Bbb R$ if and only if $\lim_{n\to\infty}\sup_{x\in A} (f_n(x)-f(x))=0$.

But, $$sup_{x\in A}\frac1{nx}=\infty$$ for all $n$.

Answer 3

Or, another way to see this, is to find a sequence of points for wich the function doesn't converge to $0$:

$\forall n\in\mathbb{N}^*, f_n(\frac{1}{n}) = 1$.

And this imply the contradiction of the uniform convergence:

$$\forall N, \exists n > N, \exists x_n=\frac{1}{n},\quad |f_n(x_n)-0|\geq 1$$