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## Homework Statement

Show that if X[itex]\subset[/itex]M and (M,d) is separable, then (X,d) is separable. [This may be a little bit trickier than it looks - E may be a countable dense subset of M with X [itex]\cap[/itex] E = Ø.]

## Homework Equations

No equations, but there are relevant definitions. Per our book:

A metric space (M,d) is separable if [itex]\exists[/itex] a countable dense E [itex]\subset[/itex] M.

E[itex]\subset[/itex]M is dense in M if [itex]\forall[/itex]x[itex]\in[/itex]M and [itex]\forall[/itex] [itex]\epsilon[/itex] > 0, [itex]\exists[/itex] e [itex]\in[/itex] E st d(x,e)< [itex]\epsilon[/itex]

## The Attempt at a Solution

My best attempt was doomed from the start, because I don't quite understand the hint. My thought process went as follows:

since X [itex]\subset[/itex] M, [itex]\forall[/itex]x[itex]\in[/itex]X, x[itex]\in[/itex]M. Thus, since M is dense in E, [itex]\forall[/itex]x[itex]\in[/itex]X, [itex]\forall[/itex][itex]\epsilon[/itex]>0, [itex]\exists[/itex]e[itex]\in[/itex]E st d(x,e)<[itex]\epsilon[/itex]. At this point, I was done, because the set of e's satisfying the above, is a subset of E, a countable set. So a subset of a countable set is dense in X, and X is separable. This is incorrect, but I cannot see why.

Any help clearing up the confusion would be greatly appreciated.

Thanks!

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