I'm checking whether I understood this problem correctly. We are to consider $$ z=\left(1,\frac12,\frac13,\cdots\right) \in (\ell^2,\lVert\cdot\rVert_2) $$ and $$ Y=(\ell^1,\lVert\cdot\rVert_2) $$ And we are to find $\mathrm{dist}(z,Y)=\inf_{y\in Y}\lVert z-y\rVert_2$.
Is it the case that the answer is 0, since we can take arbitrarily large $n$ and have $y_n\in Y$ be equal to $z$ in the first $n$ elements and 0 thereafter? I.e. $$ y_n = (1,\frac12,\frac13,\cdots,\frac1n,0,\cdots) $$
What you say is indeed completely valid. There is nothing of substance to add as what you have written is mostly complete (just the trivial calculation missing) and entirely correct. Why did you feel uncomfortable about this? Is there something I/we can clarify?