Got 1 and 2 in class. I was thinking about number 3, and got into this line of thinking. We don't know that the metric is surjective to $\mathbf{R}$, so say the metric mapped to $\mathbf{Q}$. Then certainly it could be the case that there was even a point in $X$, let alone $K$, where $d(x,k) \inf bla$. Dimberti 15:55, 21 September 2008 (UTC)

Ah, but then, just like in the discrete metric, the only compact sets would be finite ones. Nevermind. Dimberti 15:56, 21 September 2008 (UTC)

O.k., here was the line of thought I was going on, but am dubious about its success.

i) Make the argument that with the infimum you can construct a sequence that approaches it.

ii) I'm sure there's a theorem somewhere that states that point of a limit implies limit point.

iii) K is compact so it's closed.

iv) Since it's closed, the limit point is in K.

The problem I have with this is that I'm kind of meta-reading the question, so it makes the next part trivial. Dimberti 16:06, 21 September 2008 (UTC)

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Ph.D. 2007, working on developing cool imaging technologies for digital cameras, camera phones, and video surveillance cameras.