Banach/Analysis Question

Hello. I am trying to prove:

Let D be a subset of R^n+1, and let f:D->R^n be continuous. If K is a
compact subset of D, show that there exists an open U in R^n+1 such
that K is a subset of U and f is bounded on U intersect D.

I am also told to try to find a proof for an arbitrary Banach space,
rather than the real line. I didn't know what a Banach space was, so I
wiki'd it and found that it is a complete normed vector space. Since
the norm has the triangle inequality property, I assume that the norm
is the distance function for the metric. But I do not know what I
should take to be the arbitrary Banach space, the image or the preimage
of f. I also do not understand how the "completeness" part of the
Banach space is required.

I understand that if f is a continuous function on a compact set, then
f achieves its maximum and minimum and hence is bounded. But I do not
know how to extend this to an open set, as functions do not need to be
bounded on open sets.

Thanks for your help.

.

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