Request For Help With Math

[floop]

I am flipping a fair coin and counting the number of heads vs. the number of tails after each flip.

Can I prove that an infinite sequence of coin flips can always be reduced to the following:

r = f(r1) + f0 + f(!r1) + f0 + f(r1) + f0 + ...

where

r1 is the result of the first flip

f(r1) is a sequence of some number of flips (n) where the excess of heads/tails is in the same direction as r1 - that is to say if r1 is a head, f(r1) would be the sequence of flips where each count is an excess of heads, or zero.

f(!r1) is a sequence of some number of flips (n) where the excess of heads/tails is in the opposite direction as r1.

f0 is a sequence of exactly 1 flip where # heads = # tails.

Does it comes down to showing that n is always a finite number??

Is this possible? TIA.

[RobNottsUk]

Infinite number of coin flips, mean that any excess either way is infinite to.

See Hilbert's Hotel for explanation!