Originally Posted by

**godelproof** **1.** Any equivalent class contains countable elements.

Proof: Let $\displaystyle E$ be the set of all equivalent classes of $\displaystyle S$ in #7. Choose any $\displaystyle {E}_{x}\in E$ and $\displaystyle x\in {E}_{x}$.

$\displaystyle \forall y\in S$ and $\displaystyle y\sim x$ $\displaystyle \Longrightarrow$ $\displaystyle \exists {N}_{y}\in \mathbb{N}$ such that $\displaystyle {y}_{n}={x}_{n},\ \forall n>{N}_{y}$. Hence $\displaystyle y \in {\bigcup}_{k=0}^{\infty}{a}^{k}$, where $\displaystyle {a}^{0}=x$ and $\displaystyle {a}^{k}=\{({a}_{0},{a}_{1},...,{a}_{k-1},{x}_{k},{x}_{k+1},{x}_{k+2},...)|{a}_{i} \in \{0,1,...,9\}\}$ for $\displaystyle k\geq 1$. Since $\displaystyle y\sim x$ was arbitary, we have $\displaystyle {E}_{x}\subseteq {\bigcup}_{k=0}^{\infty}{a}^{k}$. Since $\displaystyle {a}^{k}$ is countable for any $\displaystyle k$, we proved that $\displaystyle {E}_{x}$ is countable. Q.E.D.