How to Prove the Cardinality of Unions of Infinite Sets?

[mufq15]

Homework Statement

Prove that the union of c sets of cardinality c has cardinality c.

Homework Equations

The Attempt at a Solution

Well, I could look for a one-to-one and onto function... maybe mapping the union of c intervaks to the reals, or something? I know how to demonstrate that a countable union of countable sets is countable, by showing how to label them.
I'm having a hard time with this one, though.

 

[AKG]

\mathbb{R}^2 = \bigcup _{r \in \mathbb{R}} (\mathbb{R} \times \{ r\} )

This should give you an easy way to associate a c-union of c-sets with R². Now all you need is a bijection between R and R².

 

[mufq15]

Ohh, I think I finally get it! (after thinking about it for a loong while...) Infinity is hard for me to wrap my head around. Thanks a lot for your help.