Let $\{ f_n \}$ be a sequence of Holder continuous functions $f_n \colon \mathbb R^d \to \mathbb R$ with Holder exponent $\alpha$ (independent of $n$) and Holder continuity constant $C_n$ such that $C_n \to \infty$ as $n \to \infty$. Is it possible that (under some conditions) $f_n$ converge uniformly on compact sets to some function $f$ (which then must be continuous) or is it not true in general?
This is of course an application of the uniform limit theorem but the fact that the (Holder) continuity estimates represented by $C_n$ blow up is it a problem for the uniform convergence? It should not as in the uniform convergence theorem the modulus of continuity of $f_n$ may depend on $n$, right?