Let $\kappa$ be an infinite cardinal. Does there always exist a function $f: \kappa \to \kappa$ such that
$f$ is weakly increasing ($f(\alpha) \leq f(\beta)$ for $\alpha \leq \beta < \kappa$)
$f$ is cofinal (equivalently, the image of $f$ is unbounded)
$f$ lies strictly below the diagnoal (i.e. $f(\alpha) < \alpha$ for all $\alpha <\kappa$)
I'm willing to relax condition (3) by allowing $f(\alpha) \leq \alpha$ whenenever $\alpha$ has uncountable cofinality.
Also, $f(\alpha)$ need only be defined for sufficiently large $\alpha$, though I'm pretty sure that doesn't make a difference.
For example, this is possible when $\kappa = \aleph_0$, by taking $f(n) = max(n-1,0)$.