PI does not contain itself

If PI contains itself then its decimal expansion is a repeating set of digits. Lets just look at the fractional part.

So pi = PI - 3. And pi(i) represents the ith digit of the fractional part of PI. 

pi = 0.141592654...

so pi(5) = 9

Now if pi contains itself then at some value of 'n' it follows that pi(i) = pi(i+n)

pi = 0.pi(1)41592654...pi(n)pi(n+1)41592654...pi(2n)pi(2n+1)41592654...

Such a sequence can be expressed as a fraction:

For example:

0.12345612345612345612345612345612... = 123456/999999

So pi would be:

1415926..pi(n)/9999999..n times

But Pi is irrational so we know that Pi is not built from a repeating finite sequence and does not repeat after a finite n.

This leaves 2 options. Either Pi repeats after a countably infinite n, or it does not contain itself.

Let us supposed that pi repeats after a countably infinite n.

This means that:

pi = lim(n->inf) 1415926..pi(n)/9999999..n

As I understand it all Real numbers can be expressed like this so the question of "contains itself" starts to get rather ambiguous. Repeating after an infinite sequence of digits is the same as not repeating it seems. To be confirmed.