Uncomputability of Busy Beaver Function https://en.wikipedia.org/wiki/Busy_beaver#Proof_for_uncomputability_of_S.28n.29_and_.CE.A3.28n.29
So this is wikipedia's proof of why Busy Beaver Function is uncomputable. But I don't get two things.

How do you know that &lt; Create_n0 | Double | EvalS | Clean &gt; has $N$ states? I understand that &lt; Create_n0 | Double | EvalS &gt; has $N$ states but I don't understand why adding Clean doesn't change the number of states.

Why do we need to make a Double function? Can't we prove like this: Let's say the Turing machine &lt; EvalS | Clean &gt; has $n_0$ states. Let $\textrm{BadS}$ denote &lt; Create_n0 | EvalS | Clean &gt;. $\mathrm{BadS}$ has $n_0$ states but has more than $S(n_0)$ shifts, therefore, contradiction.
Here's a simple proof of the non-computability of the Busy Beaver function:

Assume BB is computable. Then we can build a program that accepts a TM specification M and input x. Compute n = BB(|M|), where |M| is the number of states of TM M. Next, simulate M(x) for n time-steps. If M(x) halts, accept. If after n time-steps M(x) has not halted, reject. This program solves the Halting-Problem, a contradiction - QED.