I'm sure there's a better way to do this, but is this logically sound? (thanks for reading)

1. Let (a,b) be an open interval.

2. Assume (a,b) has finitely many points.

3. If (a,b) has finitely many points, then ∃x ∈(a,b) such that x<α ∀α ∈(a,b)

and ∃y ∈(a,b) such that y > β ∀β ∈(a,b).

4. Since (a,b) is an open interval, ∀α ∈(a,b) ∃ε such that (α - ε)∈(a,b). Likewise,

∀β ∈(a,b) ∃δ such that (β + δ)∈(a,b). Thus, ∃ε such that (x - ε)∈(a,b) and ∃δ such that (y + δ)∈(a,b).

5. But (x - ε) < x and (y + δ) > y. This contradicts 3., therefore if (a,b) is an open interval then (a,b) has infinitely many points.