Convergence depending on step

Question

Lets say I have a sequence of maps $a_n : \mathbb{R}\rightarrow\mathbb{R}$ and $a : \mathbb{R}\rightarrow\mathbb{R}$ s.t. $$\lim_{n\rightarrow\infty} a_n = a$$ uniformly. Now I now that for each $\epsilon>0$ there is a $N$ such that $\forall N\geq n$ $$\lVert a_n - a \rVert < \epsilon.$$ Can I name the $\epsilon$ explicitly depending on $n$ such as e.g. $$\lVert a_n - a \rVert < \frac{1}{n} ?$$

Answer 1

No, it is too much to ask.

For example if you let $\epsilon =1/10$ you may or may not have $||a_{10}-a||<\frac {1}{10}$ depending on your functions.

In other words we do not have control over the sequence of functions to make $||a_n-a||<\frac {1}{n}$ happen.

Answer 2

Yes you can, since $||a_n - a||< \epsilon$ holds for every $\epsilon >0$. So u can take $\epsilon = 1/n$.