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Let $X:=\{(a_n)_{n=0}^{\infty}:\sum_{n=0}^{\infty}\lvert a_n \rvert < \infty\}$ be the space of absolutely convergent sequences. Define $l^1$ and $l^{\infty}$ metrics on this space by $$ d_{l^1}((a_n)_{n=0}^{\infty},(b_n)_{n=0}^{\infty}):=\sum_{n=0}^{\infty}\lvert a_n-b_n \rvert, $$ $$ d_{l^{\infty}}((a_n)_{n=0}^{\infty},(b_n)_{n=0}^{\infty}):=\sup_{n \in \mathbb{N}}\lvert a_n-b_n \rvert. $$ To show that there exists sequence $x^{(1)},x^{(2)},\ldots$ of elements in $X$ which are convergent w.r.t. the $d_{l^{\infty}}$ metric but not w.r.t. the $d_{l^1}$ metric. I'm having trouble in constructing an example of such a sequence.

My initial attempt: Given $\epsilon>0$, we can find $N \in \mathbb{N}$ such that $\frac{1}{N}<\epsilon$ (by the archimedean property. Let $x^{(n)}=(x_0^{(n)},x_1^{(n)},\ldots)$ and $x=(x_0,x_1,\ldots)$. I'm considering $x_k^{(n)}=x_k+\frac{1}{N+k+\frac{1}{n}}$ so that the desired conditions hold (about convergence of $x^{(n)}$ to $x$ w.r.t. the two metrics on $X$). But I'm not sure about it. Any help will be much appreciated.

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    Take for example a sequence of probability distributions over $\mathbb{N}$ such that the variances are going to $\infty$. Something like the fundamental solution of the heat equation. At every time it has volume 1, but is going to 0 pointwise. – 2017-01-10
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    You want a family of sequences whose suprema get small but whose sums are bounded away from zero. Try $x_k^{(n)} = 2^{-n}$ when $k < 2^n$, $x_k^{(n)} = 0$ for $k \ge 2^n$. – 2017-01-10
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    @amcerbu: Thanks! That was helpful. – 2017-01-10

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You can construct examples consisting of finitely long sequences $x^{(n)}$. For instance, let $x^{(1)}=(1,0,0,0,\cdots)$, $x^{(2)}=(\frac{1}{2},\frac{1}{2},0,0,\cdots)$, $x^{(3)}=(\frac{1}{3},\frac{1}{3},\frac{1}{3},0,\cdots)$, $\cdots$. With respect to the $\ell^{\infty}$-metric they converge to $\mathbf{0}=(0,0,0,0,\cdots)$, but with respect to the $\ell^1$-metric they don't converge.