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?