need to prove a set of convergent sequences is closed

[gabyra]

How can I prove this?
1) Denote by c0 the subset of ?infinity consisting of all convergent sequences
x = (xi) of real numbers. Prove that the subset c0 is closed in ?infinity

2) does Q with the usual distance and The Baire space Zw contain isolated points?
I think Q does, isn't each of 1/k always isolated?.
what about the Baire space and why?
thanks for your help!

 

[pka]

I simply cannot read the awkward notation in the first question.
Please rewrite it with trying to use symbols.

For #2. Ask yourself if an isolated point can be a limit point?