real analysis - How can I prove this sequence of functions converges pointwise but not uniformly to a function?

Let $f_k(x)= \tan^{-1}(kx)$ and $f(x)=\left\{\begin{array}{rcl} \pi/2 & \mbox{if} & 0< x \leq 1, \\ 0 & \mbox{if} & x = 0. \end{array} \right.$

How can I prove that $\{f_k\}$ converges pointwise but not uniformly to $f$, wuthout using the fact that the uniform limit of a sequence of continuous functions is continuous?

I could use the theorem that says that the convergence is uniform if and only if $\lim_{k \to \infty} \sup_{x \in I}|f_k(x)-f(x)|=0$, but I don't know how to apply it.

Any hint or ideas will be very appreciated. Thank you very much.

Answer

Hint: Fix $k$. What is

$$\lim_{x \to 0^+} |f_k(x)-f(x)| \,?$$

Can you deduce from here that for all $k$ we have

$$\sup_{x \in I}|f_k(x)-f(x)| \geq \frac{\pi}{4} ?$$

Can you see how this shows that that limit cannot be 0?