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**1. Homework Statement**

Let f be a continuous mapping from metric spaces X to Y. [tex]K \subset Y[/tex]is compact. Is [tex]f^{-1}[/tex](K) bounded?

**2. Homework Equations**

Theorem 4.8 Corollary (Rudin) A mapping f of a metric space X into Y is continuous iff [tex]f^{-1}[/tex](C) is closed in X for every closed set C in Y.

**3. The Attempt at a Solution**

So my idea was to show that [tex]f^{-1}[/tex](K) was continuous, but i can't really figure that out immediately.

I just tried next to describe K and [tex]f^{-1}[/tex](K) as best I could.... We know that K is closed and compact (compact subsets of metric spaces are closed). This will imply that [tex]f^{-1}[/tex](K) is closed (Thm 4.8 corollary). So I have that K is closed and compact and that [tex]f^{-1}[/tex](K) is closed. I just don't know how to make the ends meet. Maybe I'm doing this wrong, or just missing something obvious.

Thanks in advance for any help.