Thread: help with proving a set to be compact

let K be compact in R^n. show that a + K is also compact, where a + K := { a + x : x in K}


what I was thinking is that since K is compact there exists a finite open cover of K. a is a single element and so can also be covered
by a finite open cover and the union of the two covers is also finite so a + K is contained within a finite open cover and hence compact.
am I close?

I don't see that approach working.

You could try doing it this way:



Can you show that f is continuous? If so, that's actually all you need, but you could then quickly prove much more:

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Maybe this is enough to guide you, maybe it isn't. But think about:







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You could also try doing it this way, which amounts to about the same thing while requiring more work: