# Convergence depending on step

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. $$\lVert a_n - a \rVert < \frac{1}{n} ?$$

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.

• I thought so :( Thank you :)
– Alvo
Jul 26 '19 at 16:59

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

• This holds for all $n$ exceeding a certain $N$ right?
– Alvo
Jul 26 '19 at 17:40