Question

1. Let A be a nonempty subset of R such that every number in A is greater than 2 (NOTE: This doesn’t necessarily mean that A = (2,∞)).
(a) Explain why A must have an infimum.

(b) Let c = inf(A). Prove that a∈A INTERSECTION (−∞,a] = (−∞,c].
CAN SOMEONE PLZ HELP ME WITH THIS QUESTION.

1. Let A be a nonempty subset of R such that every number in A is greater than 2 (NOTE: This doesn't necessarily mean that A = (2,0)). (3pts) (a) Explain why A must have an infimum. (b) Let c = inf(A). Prove that n(-00, a) = (-0,c). αΕΑ

Answer 1

$\small A$ is a non-empty subset of $\small \mathbb{R}$ . Also, every number in $\small A$ is greater than 2. Then 2 is a lower bound for $\small A$ . So set of all lower bounds of $\small A$ , call it L, is a non empty set. Also, the set L, is bounded by any element of A. So sup exists, and Sup(L)=inf(A).

(b) Let c=inf(A). $\small c+\frac{1}{n}>c$ . So there must exist $\small a_{n} \in A$ , such that
c+=> anc
. So we can find a cauchy sequence $\small \{a_{n}\}_{1}^{\infty} \rightarrow c$ as $\small n \rightarrow \infty$ . Now $\small \cap_{n=1}^{\infty}(-\infty, a_{n}]=(-\infty, c]$ . Also, $\small a\geq c$ , imply $\small (-\infty,a]\supseteq (-\infty, c]$ . Also, $\small \cap_{n=1}^{\infty}(-\infty, a_{n}]\supseteq \cap_{a \in A} (-\infty, a]$ . So $\small \cap_{a \in A} (-\infty, a]=(-\infty, c]$ .