$A + B = \{\,x + y \mid x ∈ A,\, y ∈ B\,\}$ is closed in $ [0,\infty)$ for any closed $A, B \subseteq [0,∞)$ If $A, B$ are closed subsets of $[0,∞)$, then
$$A + B = \{\,x + y \mid x ∈ A,\, y ∈ B\,\}$$
is closed in $ [0,\infty)$.
If $A ,B$ are closed sets in $\mathbb R$ , then i know counterexamples but considering only non negative reals ,i can't able to get a counterexample which doesn't implies it must be true.
So i want to prove or disprove , I know it is not true in genral but under this particular situation i want to know it is TRUE /FALSE

If $A$ and $B$ are compact, $A + B$ is compact because it is the image of the compact set $A \times B$ under the continuous function $(a,b) \mapsto a+b$.
If $A$ and $B$ are closed subsets of $[0,\infty)$, then for all $N > 0$ we have 
$$(A + B) \cap [0, N] = ((A \cap [0,N]) + (B \cap [0,N])) \cap [0,N]$$ which is closed, therefore $A + B$ is closed.
