Accumulation point of Cantor set

Show that the set of accumulation points of Cantor set is the Cantor set itself?

Real analysis problem with dedekind cuts

A tex-ed pdf document of the problem is attached.

Real analysis problem rational cuts in Q

A tex-ed pdf of the problem is attached. See attached file for full problem description.

Cardinality of R and R^2

How can I show that the cardinality of R and R^2, R=set of Real numbers, is equal. I think by R^2 it is meant R x R, which means an ordered pair, am I right? Is this possible just be showing that the first element in R^2 pair can be matched to R? But this is not necessarily a 'function' by definition, so there are infinite number of pairs with the same first element. How can you prove this?

Open sets - Prove that a set U subset of M is open

Prove that a set U subset of M is open if none of its points are limits of its complement. I think we may need to use epsilon-delta definitions here. help!