User blog comment:Wythagoras/Proof that TREE(n+1) divided by TREE(n) is strictly increasing/@comment-99.185.0.100-20150917002400/@comment-3427444-20150917022046

Imagine an n-state busy beaver. You can add a n+1-th state to it, and make the head move right until it finds a 0 and turns it into a 1 and halt. The original machine's instruction that leads to the halt stage is modified to lead to the n+1-th stage.

In this way after the original machine halts, the machine finds the first 0 at the right of the halting point, turns into a 1 and halts, therefore showing that BB(n+1) >= BB(n)+1 (i.e. BB(n) is strictly increasing).