I found in ‘Computational Complexity: A Modern approach’ Book the following statement that i dont quite understand its proof :

For every f : {0, 1}∗ → {0, 1} and time-constructible T : N → N, if f is computable in time T(n) by a TM M using alphabet A, then it is computable in time 4log|A|T(n)by a TM M ̃ using the alphabet {0,1,Blank,◃}.

The proof is on the link :

First Part of proof

Second Part

So basically if we have a running time of T(n) on a certain alphabet A and we wanna switch to binary.. we gonna have to multiply it by log(|A|) (log of base 2, the number of bits to represent an element from A) because every step will be multiplied by the number of bits used to represent the element.

But in the proof, starting from the second paragraph, i dont really understand what they say and how they end up with the ‘4’ in ‘4log|A|T(n)’.

Any help is welcome, Thank you so much.