Uniform limit with respect to a parameter

Question

Given a real function such that for every fixed $t\in\mathbb{R}$, $\lim_{n\rightarrow\infty}f(n,t)=0$, can I express this in English saying that $f(n,t)$ vanishes as $n\rightarrow\infty$, uniformly in $t$? My doubts are about the use of the word uniformly in the weak sense of a point-wise limit that is always the same for all $t$, when dealing with parametric limits, as it could mean something stronger, that is uniform convergence of the sequence of functions $(f_n(t))$. So in conclusion, can the adverb uniformly be used in this weak sense in English? Or it is only used for uniform limit of sequences of functions?

Answer 1

These are the formal definitions:

We say that $f_t(n)$ convergence point-wise to $0$ as $n\rightarrow\infty$ if,

$$\forall_{\varepsilon>0} \forall_{t\in\mathbb{R}}\exists_{N\in\mathbb{N}} \forall_{n>N} |f_t(n)-0|<\varepsilon$$

We say that $f_t(n)$ convergence uniformly to $0$ as $n\rightarrow\infty$ if,

$$\forall_{\varepsilon>0} \exists_{N\in\mathbb{N}} \forall_{t\in\mathbb{R}} \forall_{n>N} |f_t(n)-0|<\varepsilon$$

In other words the convergence is uniform if you can find the same $N$ for all $t\in\mathbb{R}$.

Examples:

For instance the map $f_t(n) = \frac{t}{n}$ converge point-wise to $0$ as $n\rightarrow \infty$ but not uniformly (because you need a larger $N$ the more $t$ grows).