Limit

The limit (of a Cauchy procedure) is the target which a Cauchy procedure approaches. [TODO ugh]

Examples

Planes are the limit of increasingly large spheres

Work with 3 dimensional rectilinear coordinates, and consider some sequence of spheres, $C_{1},C_{2},\cdots$ on the plane, where the sphere $C_{n}$ is centered at $(0,0,n)$ and has radius $n$. Near the origin $(0,0,0)$, the sphere $C_{n}$ will look increasingly similar to the xy plane $P:=\{(x,y,z):z=0\}$, as $n$ gets larger. For any given context, there exists some $N$ such that when $n\geq N$, the sphere $C_{n}$ is indistinguishable from the plane. We say [TODO should OM say?] that the limit of the sequence $\{C_{n}\}_{n=1}^{\infty }$ is $P$.

This example is very practical; it is why we may think of the Earth as being flat in most everyday contexts. The Earth is a sphere, but it has a radius which is so large that near a point it can be identified as a plane (or to state it more confusingly, it is a plane).

Turing Machines are the limit of increasingly robust computer programs [TODO]

Turing Machines are an idealization of computer programs [TODO no no no no no no no no no no no that's not the right way to think about it!] in the "limit" where the runtime increases. blech