This is an exercise in Durrett's probability book.
$p$ is the transition probability for a markov chain on a countable space. $f$ is said to be superharmonic if $f(x)\geq\sum_y p(x,y)f(y)$, or equivalently $f(X_n)$ is a supermartingale. Suppose $p$ is irreducible. If every nonnegative superharmonic function is constant, show that $p$ is recurrent.
It's not that easy to use the statement "every nonnegative superharmonic function is constant", so I tried 2 ways to reformulate the statement.
1.For all $f\geq 0$ nonconstant, there exists an $x$ s.t. $f(x)<\sum_y p(x,y)f(y)$. Show that $p$ is recurrent.
2.If $p$ is transient, show that there exists a nonnegative superharmonic function which is nonconstant.
But I haven't got a clue about how to prove this.