Answer to Question #174830 in Real Analysis for Aarohi

Question

Which of the following statement s are true and which are false?Justify your answer with a short proof or a counter example.

i) If x and y are real numbers such that x<y,then x^2<y^2.

ii)for every finite set S,sup S€S.

iii) There exists an interval with infimum and supremum equal.

iv)The sequence (1,1/2,1/3,1/4,....) is unbounded.

v) If a sequence is bounded ,then it has at least two convergent subsequences.

Answer 1

i) False.

$-2 <-1$ but $4 = (-2)^2 > (-1)^2 = 1$

ii) True.

Let S be a finite set. Let $x \in S$ , if x is the greatest element, then we are done. If x is not the greatest element, then $\exist y \in S$ $\ni x\leq y$ . If y is the greatest element then we are done. If y is not then, the process goes on and on. But since S is finite, we are sure this process will terminate and we can say $\exist z \in S \ni x\leq z \forall x \in S$ . This implies that z is an upper bound of S. And by the axiom of completeness, a least upper bound exist and z is the least upper bound. So we can safely conclude that the SupS is in S.

iii) False. The singleton is the only set that has equal infimum and supremum and it is not an interval.

iv) True. It does not have a lower bound.

v) False. Consider the constant sequence, say $x_n = c \forall n \in N$

The sequence is bounded but it has only one convergent subsequent.