Let $Y$ is closed subspace of $X$ and $(x_n+Y)\rightarrow (x+Y)$ in $X/Y$ for some $x_n,x\in X$.
Question is to see that there exist $y_n\in Y$ such that $x_n+y_n\rightarrow x$.
This seems to be true but i could not give a proof.
Given $\epsilon>0$ there exists $N\in \mathbb{N}$ such that $||x_n+Y-(x+Y)||<\epsilon$ for all $n\geq N$
Fix $n\geq N$ then $||(x_n-x)+Y||<\epsilon$ i.e., $\inf\{||x_n-x+y||:y\in Y\}<\epsilon$ so, I can get $y_n\in Y$ such that $||x_n-x+y_n||<\epsilon$.
I thought this sequence $(y_n)_{n\geq N}$ adding some random $y_1,\cdots,y_{N-1}$ would work. But then i see that these $y_n$ even for $n\geq N$ depends on $\epsilon$.
Suggest some hints to get rid of this $\epsilon$.
