I found this in a book : "A linear operator $T:X\rightarrow Y$ between normed linear spaces X and Y is said to be closed provided whenever $(x_n)$ is a sequence in X, if $x_n\rightarrow x$ and $T(x_n)\rightarrow y$, then $y=T(x)$."
I assumed $T$ to be a closed map (as in maps closed sets in $X$ to closed sets in $Y$) and proved the above without using the linearity of $T$. However, I am having trouble proving the converse. I assumed $A$ is closed in $X$. I have to show $T(A)$ is closed in $Y$. For that I chose a sequence $(T(x_n))$ in $T(A)$ with $T(x_n)\rightarrow y$. But I don't know if $(x_n)$ converges or not to some $x$ in order to apply the given condition.
Any help is appreciated.
Edit: The forward implication that I claimed to have proved is incorrect