How to prove by induction that a parabola corresponding to two edges intersects at atmost 2 points?

Question

I have many parabolas that are intersecting each other. I am generating a list S from the upper segments of these parabolas. Since the corresponding two edges of a parabola intersect each other at most at 2 points, the list S can contain at most 2n – 1 items.

I want to prove this by induction. What I can think of is this:

Assume I have f(x) ≤ 2n – 1.

Base case is n = 1, f(1) ≤ 2 · 1 – 1, so f(1) <= 1.

Then assume n = k holds, so f(k) ≤ 2k – 1.

We can show that for n = k+1 holds f(k+1) ≤ 2(k+1) – 1.

Am I supposed to continue like that, e.g. for n = k+2, n = k+3, …? If I continue like this, then does it mean I proved it by induction?