How do I show that an equivalence class of a language containing an empty string is infinite
Let $A$ be the alphabet. I suppose that the equivalence you are referring to is the equivalence $\sim$ defined on $A^*$ by $u \sim v$ if and only if, for all $x \in A^*$, 
$$
  ux \in L \Leftrightarrow vx \in L
$$
Now suppose there is a word $u \in A^+$ such that $u \sim \varepsilon$. Then by definition, $x \in L$ if and only if $ux \in L$. It follows by induction on $n$, that for all $n &gt; 0$, $x \in L$ if and only if $u^nx \in L$ and thus $[\varepsilon]$ contains $u^*$. If the alphabet is nonempty, it follows that $[\varepsilon]$ is infinite.