# Limit of a seqence $\{f_n \}_{n\in \mathbb N}$ of functions?

I don't really know how mathematicians talk about this concept. I try to explain better what I mean with limit of a sequence of functions:

Given a countable set of functions $\{f_n \}_{n\in \mathbb N}$ and $f_i:\mathbb R\rightarrow \mathbb R$ I call the "limit" (probably with a wrong notation) of this functions a function $\varphi$

1$\lim\limits_{n\to +\infty}f_n=\varphi$ In other words with I mean that: $\forall x\in\mathbb R\lim\limits_{n\to +\infty}f_n(x)=\varphi(x)$

This kind of "convergence" of function maybe doesn't make sense, because with numbers we can know two values are getting closer, with a set of fucntions I don't know how but I can see better this concept plotting functions: for example how the set of functions $\{t_n \}_{n\in \mathbb N}$ where $t_n(x)=^nx$ (tetration) get closer to the black line and is an example of what i mean with $t_{n\to +\infty}=\varphi$

Generalizing, what I'm intrested in is:

given a set of functions $\mathcal H:=\{h|h:X\rightarrow X\}$, a subset $\{f_n \}_{n\in \mathbb > N}\subset\mathcal H$ and $\varphi \in \mathcal H$ how can I express this:

$$\lim\limits_{n\to +\infty}f_n=\varphi$$ only if $$\forall a\in X\lim\limits_{n\to +\infty}f_n(a)=\varphi(a)$$

Questions

A) Is this the right notation? If not how/where(theory) can we talk about this concept?

B) I would like to know (if not take too much effort) if maybe which extra concepts I need for generic sets of functions $\mathcal H$

thanks in advance and sorry for grammar errors.

• Have you looked at this? en.wikipedia.org/wiki/Pointwise_convergence Apr 26, 2013 at 16:08
• When you talk about a sequence of elements of some set $X$, what you really are talking is a function from $\Bbb N$ to $X$ and the set of all such sequences/functions is usually written $\mathcal F\left(\Bbb N, X\right)$ or $X^\Bbb N$. You can't really talk about sets for sequences. From a sequence, you can get the set of all its elements but since a set isn't ordered doesn't keep count of how many of each elements there are, you can't get a sequence from a set. For example, the set $\{0,1\}$ would represent both $0,1,1,1,1,1,1,\dots$ and $0,1,0,1,0,1,0,1,\dots$. Apr 26, 2013 at 16:18
• @MphLee : Yes. Except, I wouldn't use $\{\}$ since (in my mind at least) those are for sets. I think the standard notation for sequences if what you wrote with parenthesis instead of $\{\}$. Apr 26, 2013 at 16:28
• @MphLee : Since you just called it a cartesian product, I'm not sure anymore if you understand that $\forall n \in \Bbb N, X^n \not= X^\Bbb N$. $X^\Bbb N$ isn't the first thing that comes to mind when you talk about cartesian products. It's more of a generalisation of the cartesian product. Apr 26, 2013 at 16:55
• @MphLee : I think you understood :) Even though I don't think the word tuple is appropriate for infinitely many elements. Apr 26, 2013 at 17:31

## 1 Answer

A) This concept is called "pointwise convergence" and you can read more about it on Wikipedia: http://en.wikipedia.org/wiki/Pointwise_convergence

B) You need some notion of convergence on your sets. The most general setting for this is a topological space.

• Thanks, is what I was looking for,but can you clarify a thing (in the most general scenario) the set $X$ must have a topologic structure, or the sect of functions $\mathcal H$? Apr 26, 2013 at 17:45
• If you are interested in pointwise convergence, then you are defining convergence on a sequence of functions by via convergence in the set $X$, so $X$ must have some topological structure. This induces the topological structure on $H$. There are other topological structures you could consider on $H$ -- look up "function space" or somesuch. Apr 26, 2013 at 17:59