convergences of series in two different metrics

Question

for the past few days I've been studying the topic of metric spaces and now making some exercices on it. However I'm struggling with the following exercice: let $l^1(\mathbb{N})$ denote all function $f: \mathbb{N} \rightarrow \mathbb{R}$ for which $$\sum_{k=0}^{\infty} |f(k)| < \infty$$ For $f,g \in l^1(\mathbb(\mathbb{N}))$ we look at the following metric $$d_1(f,g) = \sum_{k=0}^{\infty} |f(k)-g(k)| $$ $$d_{\infty}(f,g) = sup\{|f(k)-g(k)| | k \in \mathbb{N}\}$$ So I already proves that if $(f_n)$ is a sequence in $l^1(\mathbb{N})$ and $f \in l^1(\mathbb{N})$ that if $(f_n)$ converges to $f$ for the metric $d_1$, that is also converges for the metric $d_{\infty}$. The question now is if the reverse is also true so that if $(f_n)$ converges to $f$ for the metric $d_{\infty}$ does it also converge for the metric $d_1$. I think this is not true since taking an infinite sum over a small number gives you infinity, however I'm not able to come up with a counterexample. I also couldn't prove it. If you have any tips on how to solve this problem please let me know, any help would be grealty appreciated :))