Does this series converge pointwise or uniformly?

$\displaystyle \sum ^ \infty _ {k=1} g_k (x) = \left\{\begin{array}{cc}0,&\mbox{ if }

x\leq k\\(-1)^{k}, & \mbox{ if } x>k\end{array}\right. $

Okay, I'm terribly inexperienced with series convergence, so here is my proof after I read the chapter, please check!

I claim that this series converges pointwise but not uniformly.

Proof.

Define the partial sum $\displaystyle s_n (x)= \sum ^n _{k=1} g_k (x) $

Case 1) If $\displaystyle x \leq k $

Then we have $\displaystyle |s_n(x)-g_k(x)| = | \sum ^n_{k=0} g_k - g_k | = |0-0| = 0 $

Case 2) If $\displaystyle x>k $

Then we have $\displaystyle |s_n(x) - g_k(x) | = | \sum ^n_{k=1}(-1)^k-(-1)^k|=|-1+1+(-1)+1+(-1)+...+(-1)^n-(-1)^k|=1 $

Therefore the series $\displaystyle \sum ^ \infty _{k=1} $ converges pointwise to $\displaystyle g(x)= \left\{\begin{array}{cc}0,&\mbox{ if }

x\leq k\\1, & \mbox{ if } x>k\end{array}\right. $

I know that I still need to prove this thing is not uniformly convergence, but am I on the right track so far? Thanks!