6. In each of the following three subquestions, for each of the three statements indicate whether it is
true or whether it is false. You do not need to justify your answers. If you don't know an answer and
guess, you won't be penalized.
(a) (i) Every set of real numbers has a least upper bound.
(ii) If a set of real numbers is bounded, then it has a maximum.
(iii) If a set of real numbers is not bounded above, then it has no least upper bound.
(b) (i) Every decreasing sequence that is bounded above is convergent.
(ii) Let (xn) be a sequence such that for all real numbers ε > 0 there exists a positive
integer N such that
for all n ≥ N. Then
(iii) Every convergent sequence is bounded above.
(c) (i) If , then the series ∑
(ii) If (xn) is a Cauchy sequence, then (- xn) converges.
(iii) We have | | | | | | for all x, y ∈ R.
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