# Distance from a closed set.

Let $$X$$ be a normed linear space and let $$Y$$ be a closed subspace of $$X$$. Suppose $$(y_n)$$ be a sequence in $$Y$$ such that $$d(y_n, A)\to 0$$ for a subset $$A$$ of $$X$$. Is it true that $$d(y_n, A\cap Y)\to 0$$? Neither I could prove nor could I find a counterexample. Any hint is appreciated.

Suppose $$X=\mathbb{R}^2$$ with the Euclidean norm. Let $$Y=\mathbb{R}\times\{0\}$$ and $$A=\{(0,\frac{1}{n}):n\in\mathbb{N}\}$$.
Now take $$y_n = (\frac{1}{n},0)$$ then $$d(y_n,A) = 0$$ because $$y_n$$ is arbitrarily close to $$(0,0)$$ which is arbitrarily closed to an element in $$A$$. But $$A\cap Y=\emptyset$$ so $$d(y_n,Y\cap A)$$ is not even defined. (If you want that $$d(y_n,Y\cap A)$$ will be defined just add any element that is not $$(0,0)$$ to $$A$$).
• I am sorry I didn't mention that $A\cap Y\neq \emptyset$. – Anupam Jun 9 '19 at 19:01
• @Anupam Just add $(2,0)$ to $A$ and that's it. – Mark Jun 9 '19 at 19:03