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**Pinkk** **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 $\displaystyle \{x_{n}\}$ is a sequence in $\displaystyle S$, then it has a subsequence $\displaystyle \{x_{n_{k}}\}$that converges to some $\displaystyle x\in \mathbb{R}^{n}$, and since $\displaystyle f$ is continuous there $\displaystyle \{f(x_{n_{k}})\}$ is a convergent sequence in $\displaystyle f(S)$, which means that sequence is bounded. Now I am not sure if I can conclude that $\displaystyle f(S)$ 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 $\displaystyle \mathbb{R}^{n}$, is bounded, then $\displaystyle f(S)$ is bounded? Any help would be appreciated, thanks.