Sum of connected subsets is connected

Question

Let $A,B$ be connected subsets of the Real Numbers. Show that $A+B$ is a connected subset of the Real Numbers.

Answer 1

If you don't want to use that every connected set in $\Bbb R$ must be an interval, you can use that $A \times B \subseteq \Bbb R^2$ is connected, and $A+B = f(A\times B)$, where $f\colon A \times B \to \Bbb R$ is given by $f(a,b) = a+b$ (and we're using that continuous images of connected spaces are again connected). This works for arbitrary normed spaces.