Exercise 6.45. Let E be a nonempty open set in R. For each x € E, let Ix = (ax,bx), where ax = inf {a ER: (a,x) CE} and b = sup {b R: (x, b) CE}. (i) Prove that E = Ux€E Ix. TEE (ii) Prove that if x € E, y € E, and IxIy ‡ Ø, then Ix = Iy. (iii) Prove that the collection U = {I: x € E} is countable.

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A: Explanation of the answer is as follows

Q: 1. (a) Consider the set S = {n(-1)" : n e N}. Verify the existence or otherwise of Min(S) and…

A: a) The given set is,S=n-1n:n∈ℕ

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