This is homework and I'm not looking for an answer. I just finished an exercise that asked to prove that every metric compact space can be embedded in the Hilbert cube. Knowing this I can see that I have to find a non-compact metric space to start, but I don't think that's enough because I've been thinking for a while now and nothing has come to mind, so a little intuition or hints on what I'm looking for would be appreciated. Thanks.
HINT: It isn’t really compactness that matters here: it’s second countability. You want a metric space that isn’t second countable; equivalently, you want one that isn’t separable. There are some really simple non-separable metric spaces. A further hint is in the spoiler-protected block below.
Try discrete spaces.