# Set of surjective bounded linear operator between two Banach spaces.

I've to prove that the set of all subjective bounded linear operator is a closed subset of B(X, Y).

I tried with a sequence $$T_n$$ of surjective bounded maps, suppose it converges to some T $$\in$$ B(X, Y).

I've to show now T is surjective and bounded.

Let y $$\in$$ Y, for each $$T_n$$ there exists $$x_n \in$$ X such that $$T_n(x_n)=y$$.

Can someone help me how to proceed from here?

This is false. Take $$Y$$ to be the scalar field. Let $$f$$ be any non-zero continuous linear functional on $$X$$ And $$f_n=\frac f n$$. Then $$f_n \to 0$$ in operator norm and each $$f_n$$ is surjective.