## Chapter 6.5:

## Terras Theorem

On one particularly demoralized day working on this impossible math problem, in frustration I threw aside my own papers and journals off the table and decided to look up: Okay, what do we know about Collatz Space? I decided to go all the way back to where I had started (covered in Chapter 1.4) and take a look at all the known results ever proven in Collatz Space. I watched lectures, pulled dusty tomes off library shelves... and then one day my jaw absolutely hit the floor when I reencountered this theorem (I had read it once years before):

Theorem (Terras):

Let M be a positive integer.

Let D(M) be the fraction of numbers < M that do not have finite stopping time.

Then the limit of D(M) for M → ∞ is equal to 0.

Proof:

Found on this website:

Do you see the connection?...