  We prove that if $\Sigma_{\mathbf A}(\mathbb N)$ is an irreducible Markov
shift space over $\mathbb N$ and $f:\Sigma_{\mathbf A}(\mathbb N) \rightarrow
\mathbb R$ is coercive with bounded variation then there exists a maximizing
probability measure for f, whose support lies on a Markov subshift over a
finite alphabet. Furthermore, the support of any maximizing measure is
contained in this same compact subshift. To the best of our knowledge, this is
the first proof beyond the finitely primitive case on the general irreducible
non-compact setting. It's also noteworthy that our technique works for the full
shift over positive real sequences.
