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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 $\epsilon$.

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For each $n$, you can find $y_n\in Y$ with $$\Vert x_n-x+y_n\Vert\leq\Vert (x_n+Y)-(x+Y)\Vert+1/n$$ by definition of the norm on the quotient space.

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