Let $A$ be a closed, convex, set in a Banach space $X$, and let $B$ be a closed, bounded, convex set in $X$. Assume that $A \cap B = \emptyset$. Set $C = A- B$. Prove that $C$ is closed, and convex.
So proving $C$ is convex is not too hard, however I am having issues proving it is closed. According to existing convex geometry literature, this is a well-known result, that requires the assumption of $B$ being bounded. However, I am not sure how to use it. I assume the proof would begin by picking $c_n \in C$ such that $c_n \to c$, $c \in X$, and showing that in fact that $c \in C$. We would write: $$ c_n = a_n - b_n $$ and somehow show that $c = a -b$ for some $a,b \in C$, presumably $a = \lim_{n \to \infty} a_n$, $b = \lim_{n \to \infty} b_n$. But the existence of the limit for $c$ says nothing about the limits of $a_n,b_n$ (it can be deduced that $a_n$ is norm bounded using the norm bound on $b_n$), but I am truly at a loss on how to proceed. A hint would be appreciated.