Let $X$ be a Hausdorff topological vector space and $N$ be a closed subspace. Consider the quotient topological vector space $X/N$ with natural quotient topology. I want to show the following statement:
If $x_n+N\to x+N$ in $X/N$, then for any representatives $x_n,x$, there exist $y_n\in N$ so that $x_n+y_n\to x$.
My idea for proof is motivated by the simple example: $X=\mathbb R^2$ and $N=\{0\}\times \mathbb R$, in which the result is simple. But how could I proceed in infinite dimensional space?