Let $M$ is a closed subspace of a normed space $X$ $x_n+M \to x+M$ in $X/M$ if and only if there exists $(y_n)_{n\in \mathbb N} \in M$ such that $x_n+y_n \to x$
Since M is closed $\forall y \in M$ $\exists y_n \in M$ such that $y_n \to y$ (wrt norm of $X$ space)
If $x_n \in X$ and $y_n \in M $ then $x_n+y_n \in X/M$
I cannot think anything about convergences. I really confused and I need an elementary proof or hints for getting rid of this confusion. In addition, I’ve seen it as a necessary condition in some sources. Is it sufficient? Very very thanks in advance
$x_n+M \to x+M$ iff $||(x_n+M)-(x+M)|| \to 0$. But $||(x_n+M)-(x+M)||=||(x_n-x+M)||=\inf_{y \in M} ||x_n-x+y||$. If there exists $\{y_n\} \subset M$ such that $x_n+y_n \to x$ then $\inf_{y \in M} ||x_n-x+y|| \leq ||x_n-x+y_n|| \to 0$. Conversely, suppose $\inf_{y \in M} ||x_n-x+y|| \to 0$. Then $\inf_{y \in M} ||x_n-x+y||+\frac 1 n>||x_n-x+y_n|| $ for some $y_n \in M$. It follows that $x_n+y_n \to x$.