Let $A_1, A_2$ be closed subspaces of a Banach space $B$ such that $A_1 \cap A_2 = \{0\}$ and:
$\inf\{\|x-y\|:x\in A_{1}, y\in A_{2}, \|x\|=\|y\|=1\}>0$
Show that $A_{1}+A_{2}$ is closed.
I'm completely lost on how to use the infimum assumption given. I know of a similar result for metric spaces, in where if we have a compact set $A$ and a closed set $B$ with trivial intersection, then we must have that $d(A,B)>0$. As well, I'm not sure on how to use the fact that the space is Banach. I was thinking of taking a sequence $x_n+y_n \to z$ and then showing $z \in A_{1}+A_{2}$, first by showing that the component sequences $x_n$ and $y_n$ are Cauchy, and then using completeness (i.e. closed subsets of a complete vector space) to find that $z=x+y$ for some $x\in A_1, y\in A_2$.
What would be the best/simpler way to solve this problem, preferably avoiding quotient spaces? Thanks for the help.