# Direct proof to show that a set is closed

I was trying to prove something, and I did it, but what I used is too exaggerated. The problem is:

Let K be the cantor set, prove that the sets \eqalign{ & \left\{ {\left| {x - y} \right|\,:x,y \in K} \right\} \cr & \left\{ {x + y\,\,:\,x,y \in K} \right\} \cr} are closed in the real numbers.

What I did is said that these functions are continuous, and because $K \times K$ is compact then its image is also compact, and so the set must be closed in $\mathbb{R}$. Can I prove this directly?

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Your proof is a beautiful one-liner. Why would you want something else? A "more direct" proof will likely just explicitly reprove the facts you've used. –  Anton Geraschenko Aug 23 '11 at 18:18
Like Anton said. Or, prove that the first set is $[0,1]$ and the second one is $[0,2]$. –  Did Aug 23 '11 at 19:25
How is using basic facts of point-set topology too exaggerated? –  JSchlather Aug 23 '11 at 19:53