I have a language $\mathrm{Count}(M)$, defined below, and a finite number $k$.
\begin{align} \mathrm{Count}(M)= \{k \in \mathbb{N} \mid \text{there exists some input on which $M$ halts after exactly $k$ steps}\}. \end{align}
I need to check whether the language $L$ is in RE, and whether it is in R. $$L= \{ \langle M \rangle \mid \text{$\mathrm{Count}(M)$ is an infinite set}\}.$$
My idea for a solution starts with: even though $L$ is an infinite language, it contains $\mathrm{Count}(M_i)$ that halts eventually. Each $M$ in $L$ must halt, and $\mathrm{Count}(M)$ is infinite.
So, in order to prove that $L$ is in RE, can I simply say every $M$ in $L$ halts eventually?