Excellent - you've saved me the effort of doing it.
I note that your proof states an assumption "and also the resulting 2n pieces are distributed as evenly as possible between the n-sticks (i.e. every n-stick is composed of at least ⌊2n/m⌋ and at most ⌈2n/m⌉ pieces)".
I don't think this is required to be an assumption - if any n-stick is composed of fewer than ⌊2n/m⌋ pieces, the bound holds trivially, and n-sticks consisting of more than than ⌈2n/m⌉ pieces cannot result in there being fewer n-sticks made out of ⌊2n/m⌋ pieces (and the number of n-sticks made out of ⌊2n/m⌋ pieces is what the proof depends on).
no subject
I note that your proof states an assumption "and also the resulting 2n pieces are distributed as evenly as possible between the n-sticks (i.e. every n-stick is composed of at least ⌊2n/m⌋ and at most ⌈2n/m⌉ pieces)".
I don't think this is required to be an assumption - if any n-stick is composed of fewer than ⌊2n/m⌋ pieces, the bound holds trivially, and n-sticks consisting of more than than ⌈2n/m⌉ pieces cannot result in there being fewer n-sticks made out of ⌊2n/m⌋ pieces (and the number of n-sticks made out of ⌊2n/m⌋ pieces is what the proof depends on).