Abel's theorem infinite case

I would like to prove the second remark in the Wikipedia article of Abel's theorem:

Let $a_k $ be real numbers. If $\sum_{k=0}^{\infty} a_k = +\infty $ then $$\lim_{z\to 1^-} \sum_{k=0}^{\infty} a_k z^k =+\infty, $$provided that the radius of convergence of the series is $1$.

I tried to use the same ideas as in the usual case where $\sum_{k=0}^{\infty} a_k<\infty$ but without success.

Any help would be appreciated. Thank you.