# Convergence ### Functional Sequences We generally start our journey of sequences as they pertain to *sequences of numbers*, such as the example below:  ### Pointwise Convergence Now, what does a sequence of functions look like: $(f_n)_{n \in \mathbb{N}} = (f_1, f_2, f_3, \dots)$ A functional sequence is an infinite list of functions which we call $f_1, f_2, f_3, \dots$, where each element in our sequence is a *function*, not a number. So, to illustrate a concrete example, we may have the sequence of functions: $f_1(x) = x, \;\; f_2(x) = \frac{x}{2}, \;\;f_3(x) = \frac{x}{3}, \dots$ In this case, the general rule for our sequence is: $f_n(x) = \frac{x}{n}, \;\; \forall \;\; (n \in \mathbb{n}, x \in \mathbb{R})$ We can visualize the limiting behavior of our sequence of functions as follows:  Our functions will get closer and closer to matching the *zero function*. Now suppose that we look at the fixed value of $x=1$. We see that the function values at $x=1$ converge to $0$. Likewise, the same occurs for $x=2$. They simply occur at a different speed.  The important point is: > If we choose any *fixed value* of $x$, the points $f_n(x)$ converge to $0$. This is also the value of the limit function, $f(x) = 0$.  To summarize, suppose we have a fixed value of $x \in \mathbb{R}$, then: $\frac{x}{n} \rightarrow 0 \; \text{as } \; n \rightarrow \infty$ We then say that the functional sequence $\big( \frac{x}{n} \big)_{n \in \mathbb{N}}$ **converges pointwise** to the limit function $f(x) = 0$. It is called *pointwise convergence* because it describes what happens at fixed points, in other words fixed values of $x$. We cannot forget that the limit is a function and not a number. When we say that $\frac{x}{n}$ converges to $0$, we are talking about the constant function $f(x) = 0$ and not the number $0$.  Note that in the above definition, $N$ can depend on both $\epsilon$ *and* $x$. This will not be the case for uniform convergence. ### Uniform Convergence Now suppose that we have a number $\epsilon$, and that it is *fixed* and *small*. $\epsilon$ in our diagram represents a certain distance from our limit function, $f(x)=0$.  We can ask ourselves the question: How large does $n$ need to be (how far do we need to go in our sequence) for: $|f_n(x) - f(x)| < \epsilon$ The answer to this question actually depends on what value of $x$ we are looking at! For example, consider the following values of $x$:  We see that for the first (left) value of $x$, $n=3$ is large enough to satisfy our inequality. However, for the second (right) value of $x$, we would need $n \geq 4$. **Key Idea** > Because the value of $n$ required for our inequality to hold actually depends on $x$, we say that our functional sequence is **pointwise convergent**, but *not* **uniformly convergent**.  The difference between this definition and that for pointwise convergence is that our quantifiers have swapped around and are now in a different order. Here, $N$ can depend only on $\epsilon$, *not* on $x$. In other words, the same value of $N$ has to work for all values of $x$. ### Comparison of Pointwise vs. Uniform Convergence  --- Links: [Mathematics MOC](Mathematics%20MOC.md) References: * https://www.youtube.com/watch?v=l6v8ozOi1lg