You should be able to prove it by showing that the existence of a faster-growing computable function lets you solve the halting problem.

Assume f(x) is computable and grows faster than BB(x). We can create a computable g(x) where g(x) > BB(x) for all x, by setting g(x) = f(x) + C for some suitable constant.

Now we create a computer program that does the following. 1. Get the size of the input program and compute g(n). 2. Run the input program for g(n) steps. 3. If it stopped, return true. If it's still running, return false.

This solves the halting problem. Since a computer program can't solve the halting problem, this program can't exist. Everything except "compute g(n)" is straightforward, so there must be no such function g, and no such function f.