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Suppose $ Z_n$ be a sequence in $A+B$ and assume that it converges to some point $Z$. We have to show that $z\in {A+B}$Now $x_n\in A $ and $y_n\in B$ then $z_n=x_n+y_n$ Now as $x_n\le x_n+y_n$ and $y_n\le x_n+y_n$ so $x_n, y_n$ are bounded.And as $A$ is closed so exists a subsequence$x_{n_i} \to x$ as$ i\to \infty$ and $x\in A$Thus $y_{n_i}=z_{n_i}-x_{n_i}\to z-x$As $B$ is closed so $z-x\in B$ So $z=x+(z-x)$ is in $A+B$

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