| Putting that into plain English, "If it looks like a sequence is converging, it really is converging to something." What does it mean to say that the sequence looks like it is converging? Naively, it means that any two elements far in the sequence are always very close together. We make that intuition more precise by turning it into a challenge-response, "You tell me how close you want the elements to be, I'll tell you how far out to pick your elements." And then we write that mathematically as ∀ ϵ > 0 # You tell me how close
∃ N # I'll tell you how far
such that if N < n, m # so that any 2 elements
then || a_n - a_m || < ϵ # will always be that close
Others have given examples about things like the rational numbers. The canonical example of why it matters comes from Fourier series. Joseph Fourier discovered this one. Consider a bunch of functions f(x) over the interval from 0 to 2π. We can create a dot product with f·g equaling the integral from 0 to 2π of f(x)g(x). And the length of a function f is sqrt(f·f)Joseph Fourier discovered that the following functions are orthonormal (each has length 1, their dot product with each other is 0). 1/sqrt(2π),
sin(x)/sqrt(π), sin(2x)/sqrt(π), sin(3x)/sqrt(π), ...
cos(x)/sqrt(π), cos(2x)/sqrt(π), cos(3x)/sqrt(π), ...
(I hope I have the constants correct...)Given any function, he could compute a series that we now call the Fourier series that looks like it added up to that function. But there were complications. It added up perfectly for smooth functions with the same value at the edges like x(2π-x). But it also added up except at a few points for things like square waves. This caused a crisis in mathematics because at the time square waves were not considered functions, and nobody had ever realized that adding up infinite series of smooth functions could do such weird things. Their idea of functions was essentially what we would call analytic functions today. Basically things that look like power series. And Mr. Fourier had just shown that the analytic functions are not complete. In addition to revealing problems in how we understood math, his technique was very, very useful. Because now you just had to figure out the physics of how, say, heat spreads out or vibrating strings vibrate just for those those sin and cos terms, and then you could figure out heat and vibration for ANY function. Resolving the math problems started many decades of research. The results of which included better definitions of the real numbers, the ϵ-δ definition of a limit (or ϵ-N for a series), new theories of integration, and the idea of Hilbert spaces. |