Pseudo-uniform convergence, a nonstandard treatment

We introduce and study the notion of pseudo-uniform convergence which is a weaker variant of quasi-uniform convergence. Applications include the following nonstandard characterization of weak convergence. Let $X$ be an infinite set, $B(X)$ the Banach space of all bounded real-valued functions on $X,$ $\{f_{n}: n\in N\}$ a bounded sequence in $B(X),$ and $f\in B(X).$ Then the sequence converges weakly to $f$ if and only if the convergence is pointwise on $X$ and, for each strictly increasing function $\sigma :N\to N$, each $x\in ^{*}X$, and each $n\in ^{*}N_{\infty }$, there is an unlimited $m\leq n$ such that $^{*}f_{ ^{*}\sigma (m)}(x) \simeq ^{*}f(x)$.