# Uniform convergence as $\epsilon\to 0^+$

Reading some lectures on Hamilton-Jacobi PDE theory I found some terminology that I really don't understand.

Let $$\Omega$$ be an open subset of $$\mathbb{R}^n$$. Suppose that $$u_\epsilon:\Omega\to \mathbb{R}$$ is a function for $$\epsilon>0$$. Consider a sequence of function $$(u_\epsilon)$$ and let $$u:\Omega\to\mathbb{R}$$.

What does $$u_\epsilon$$ converges uniformly to $$u$$ as $$\epsilon\to 0^+$$ means?

I mean, I know what uniformly convergence means for a sequence of functions $$(f_n)$$ as $$n\to +\infty$$, but I really don't understand this terminology.

Thanks a lot in advance.

Let $$\{u_h\}_h$$ be a family of functions indexed by $$h\in S\subset (0,\infty)$$. We say that $$u_h\to u$$ uniformly as $$h\to 0^+$$ if for a given $$\varepsilon>0$$, we can find $$\delta>0$$ such that $$\sup_{x\in\Omega}|u_h(x)-u(x)| < \varepsilon$$ for all $$h\in S\cap(0,\delta)$$.
Remark: Using this definition, we can see that $$u_n\to u$$ uniformly in the usual sense iff $$\hat u_h\to u$$ uniformly as $$h\to 0^+$$, provided that we take our index set to be $$S=\{1,\frac 12,\frac 13,\dots\}$$ and $$\hat u_{1/n} := u_n.$$