Show that the set

is equivalent to

. (From lecture, we know that

is uncountable. The question is how to put this set in a 1-1 correspondence with

.)

\emph{Proof.} Let

be any real number and

a sequence that converges to x in

such that the ith element of

is the ith decimal approximation to x. Now, consider the decimal expansion of

, and to each

correspond the set

Now, letting

we have that our x corresponds to the set

. Since x was arbitrary, this holds for any real number

. Thus, we have established a one-to-one correspondence

, and therefore,

is uncountable.

Is this proof correct? Do I really need a converging sequence to x?

And how do I get LaTeX to show up here?