If f: R^n --> R^m is continuous everywhere and S is bounded, then f(S) is bounded
I have a general idea on how to go about it; if
is a sequence in
, then it has a subsequence
that converges to some
, and since
is continuous there
is a convergent sequence in
, which means that sequence is bounded. Now I am not sure if I can conclude that
is bounded. If I can't, how would I go about proving that if a function is continuous everywhere and a set S, subset of
, is bounded, then
is bounded? Any help would be appreciated, thanks.