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**davismj** Show that the set $\displaystyle $P(\mathbb{N})$$ is equivalent to $\displaystyle $\mathbb{R}$$. (From lecture, we know that $\displaystyle $P(\mathbb{N})$$ is uncountable. The question is how to put this set in a 1-1 correspondence with$\displaystyle $\mathbb{R}$$.)

\emph{Proof.} Let $\displaystyle $x \in \mathbb{R}$$ be any real number and $\displaystyle ${x_n} \rightarrow x$$ a sequence that converges to x in $\displaystyle $\mathbb{R}$$ such that the ith element of $\displaystyle ${x_n}$$ is the ith decimal approximation to x. Now, consider the decimal expansion of $\displaystyle $x_i = d_0 + d_1(10)^{-1} + \ldots + d_i(10)^{-i}$$, and to each $\displaystyle $x_i$$ correspond the set $\displaystyle $\{{d_0},{d_1(10)},\ldots,{(10^i)d_i}\}$. \\$

Now, letting $\displaystyle $i \rightarrow \infty$,$ we have that our x corresponds to the set$\displaystyle $\{{d_0},{d_1(10)},\ldots\} \in P(\mathbb{N})$$. Since x was arbitrary, this holds for any real number $\displaystyle $x \in \mathbb{R}$$. Thus, we have established a one-to-one correspondence $\displaystyle $\mathbb{R} \mapsto P(\mathbb{N})$$, and therefore, $\displaystyle $P(\mathbb{N})$$ 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?