Define Jn := (0, 1/n) for n ∈ N. Prove that (Jn)n≥1 is nested and∩n≥1Jn = ∅

Step 1: Show that (Jn)n≥1 is nested

The sequence (Jn)n≥1 is said to be nested if for every m, n ∈ N with m > n, we have Jm ⊆ Jn.

Let's take any m, n ∈ N such that m > n. Then, 1/m < 1/n.

So, for any x ∈ Jm, we have 0 < x < 1/m < 1/n.

This implies that x ∈ Jn.

Therefore, Jm ⊆ Jn.

So, the sequence (Jn)n≥1 is nested.

Step 2: Prove that ∩n≥1Jn = ∅

The intersection of all Jn, denoted by ∩n≥1Jn, is the set of all x such that x ∈ Jn for all n ∈ N.

We need to show that this set is empty, i.e., there is no x that belongs to every Jn.

Suppose, for contradiction, that there exists an x such that x ∈ Jn for all n ∈ N.

Then, for each n ∈ N, we have 0 < x < 1/n.

But as n approaches infinity, 1/n approaches 0.

So, we have 0 < x < 0, which is a contradiction.

Therefore, our assumption that such an x exists is false.

Hence, ∩n≥1Jn = ∅.

This completes the proof.