1. The following proof has an error. Find the error that makes the proof incorrect.
Claim 1. The set [0, 1] has no least upper bound.
Proof ”. For sake of contradiction, let a = sup[0, 1].
Case 1, If a>1, then for every x∈[0,1], it is true that x<a. Therefore a is an upper bound of [0, 1].
However 1 < (1 + a)/2 < a,
so by letting b = 1+a , we have another upper bound of [0, 1]. Since b < a, it follows that a cannot be a least upper bound of [0, 1].
Case 2, If a ≤ 1, then no matter the value of a, there is at least one x ∈ [0,1] that makes the statement x < a false (take x = 1, for instance).
Thus a does not fit the definition of upper bound of the set [0, 1].
Hence a is not a least upper bound, in particular.
Therefore, for every a ∈ R, either a is an upper bound for [0, 1] but not a least upper bound, or that a is not an upper bound of [0,1] at all.
Therefore sup[0,1] does not exist.
2. Can the error be fixed above? If so, give a correct proof of the claim. If not, explain why the claim is incorrect, giving a proof or counterexample, if needed.