Intuition behind defining convergence of functions What is the intuition behind defining convergence of sequence of functions? I didn't get a satisfying answer.
 A: There are two kinds of convergence you should definitely understand: pointwise and uniform convergence. Call the sequence of functions we are looking at $\{f_n\}$, and we'll assume the functions are $\mathbb{R} \rightarrow \mathbb{R}$ for convenience.
For pointwise, pick a point $x$ in $\mathbb{R}$, and define the sequence $a_n = f_n(x)$. If the sequence converges pointwise to a function $f$, then this sequence will converge to $f(x)$. There is no condition about how fast it converges, but given some $\varepsilon > 0$, we have to be able to pick $N \in \mathbb{N}$ such that for all $n > N$, $\lvert f(x) - f_n(x) \rvert < \varepsilon.$
On the other hand, if we want uniform convergence, we are given $\varepsilon > 0$, and we then have to be able to pick an $N \in \mathbb{N}$ such that given a point $x$, the values of $f_n(x)$ for $n > N$ are all within $\varepsilon$ of $f(x)$ (i.e. $\lvert f(x) - f_n(x) \rvert < \varepsilon$). This should look very similar to the other condition! I got them mixed up a lot in real analysis. The distinction is that for uniform, you pick $\varepsilon$ and $N$ first, and for pointwise, you pick the point first. This is an important distinction: if you pick $\varepsilon$ first and need to find an $N$ that works for all points $x$, it is possible that you cannot -- even if the functions converge pointwise, some points may require larger $N$s than others, and the size of these $N$s might go to $\infty$.
Uniform convergence is stronger than pointwise, and therefore it is more powerful. For example, if $\{f_n\}$ converges uniformly to $f$, then $\int f_n$ converges to $\int f$. This is used very often when we want to approximate a function with functions that are somehow easier to work with. For example, if every function $f_n$ is a simple function, one which is made of discrete steps, it is very easy to integrate (you just add up the area under each step). Every continuous function can be defined as the uniform limit of a sequence of simple functions, so we can find the integral by taking the limit of the integrals of these. For example, if we are approximating the integral of the parabola $x^2$, $\int f_1$ might look like the shaded area in this image, $\int f_2$ like this, etc. The simple functions will converge uniformly to $x^2$, which means that $\int f_n$
will converge to $\int x^2$.
A: The biggest leap is that between pointwise and uniform convergence. Pointwise convergence tells you basically nothing, just that for each $x$, the sequence $f_n(x)$ converges. Since in this sense, all the $x$'s are totally separate from one another, there's no reason to think this mode of convergence tels you anything at all.
Uniform convergence on the other hand, has a nice interpretation once you understand a little bit about topology. You may consider the space of continuous functions on a compact topological space $X$ with values in $\mathbb{R}$ and denote this space by $C(X)$. We can topologize this space by in fact giving it a metric, a standard metric being $\rho (f,g) = \sup_{x \in X} |f(x) - g(x)|$. This lets us regard the functions as points in a topological space. Since $f$ and $g$ are continuous on a compact set, they are bounded and so this quantity is finite. Alternatively you could drop the condition that $X$ be compact and restrict yourself to bounded functions, or compactly supported functions.... you get the point.
Anyway, it's worth noting that if you have a sequence of functions $f_n \in C(X)$, this being a perfectly good metric space, we can talk about convergence of functions $f_n \to f$. This will depend on the topology of the space in general, but for our particular metric, it merely says that if $\epsilon$ is given, there is $N$ such that $\sup_{x \in X} |f_n(x) - f(x)| < \epsilon$ when $n \geq N$. But this is exactly what it means for convergence of the $f_n$ to be uniform.
Whenever a notion of convergence is defined, it will be of this form. There is always some appropriate underlying topological space of functions with a choice of metric that makes this concept of convergence equivalent to the concept of convergence of a sequence of points in this topological space.
Many theorems of analysis are made more enlightening (by my view) with this idea in mind. The Bolzano-Weierstrass theorem asserts that bounded sequences contain convergent subsequences. Thinking of a sequence of points as a sequence of constant functions, we can see this is a special case of the Arzela-Ascoli theorem, which says that a sequence of uniformly bounded and equicontinuous functions has a uniformly convergent subsequence. This follows since a bounded sequence of constant functions is uniformly bounded, equicontinuity for constants is trivial, and convergence is uniform because the functions are again, constants.
A: There are several definitions of the convergence of sequences of functions. And in some sense every definition relies on a different intuitive idea of convergence. 
Given a sequence of functions $f_1,f_2,\ldots$, for instance functions
$$f_n\colon [0 , 1] \to \mathbb{R}$$
one intuition that could work for many definitions is that of thinking of the sequence of graphs of the $f_n$ and "see" if they tend to some other graph that could be thought of as being the graph of a certain function $f$. In that case, you would like to be able to say that $f_n \to f$ as $n \to \infty$.
But of course, there's no obvious definition of what it is for a "sequence of graphs" to tend to another graph. For instance, if you think of the graphs (in $[0,1]$ of the functions $f_1(x)=\tfrac 1 2 x$, $f_2(x)=\tfrac 2 3 x$ and, in the general case, $f_n(x)=\tfrac n {n+1} x$, you would find it more and more "close to" the graph of $f(x)=x$.
This "closeness" happens, for many reasons:


*

*for a fact, if you think of $x$ as fixed number in the domain and take the limit as $n$ tends to $\infty$, you will find the answer is $x$ in every case;

*if you think of the functions $g(x)=x+\epsilon$ and $h(x)=x-\epsilon$, you will also find that for every $\epsilon>0$, there exists a natural number $N$ such that $f_n(x)$ "lies between" $g(x)$ and $h(X)$ (that is, $h(x)\leq f_n(x) \leq g(x)$ for all $x\in [0.1]$) given that $n$ be greater than $N$. Graphically, you would see that the graphs get inside completely between two copies of the graph of $f(x)$: one displaced $\epsilon$ units up, and another moved down; even for very small epsilons, there is a certain $N$ such that the graphs of $f_{N+1}, f_{N+2},\ldots$ all lay completely between the two copies of that of $f$.


The first notion leads to point-wise convergence while the second one leads to uniform convergence (AKA, convergence in infinity norm). Many others related to the "graph intuition" can be defined and perhaps others that do not relate to it but for sure are based on some other intuition.
