The logic of the proof is as follows:
To show completeness, you require that every Cauchy sequence converges.
To be a Cauchy sequence, you require a condition to hold for all choices of
ϵ.
If you choose
ϵ to be 1/2 (or any positive value less than one), you note that this implies that the sequence is eventually constant.
An eventually constant sequence is a convergent sequence.
Think of the condition to be a Cauchy sequence as a "challenge/response" condition. Given a sequence, and my choice of
ϵ does it obey the rule? If you choose
ϵ≥1 then any sequence obeys the rule; but this is not enough, as the sequence has to obey the rule for any choice of
ϵ.