Restricted Jordan curve theorem

[ehrenfest]

Homework Statement

http://ieeexplore.ieee.org/iel5/9518/30166/01385873.pdf?arnumber=1385873

I am reading a proof of the Restricted Jordan curve theorem (see Lemma 2 of the link) and the first two sentences are:

"Because the list of segments is finite, nonintersecting segments cannot be arbitrarily close. Hence we can leave a face only by crossing C."

I do not understand why the second sentence follows from the first.

Homework Equations

The Attempt at a Solution

 

[EnumaElish]

I could not get to Lemma 2, only the abstract.

 

[ehrenfest]

What do you mean? It is on the top of page 6.

 

[morphism]

You probably need a subscription to access that page (and such a subscription would likely be available through a university connection).

 

[ehrenfest]

Oh yeah. I forgot about the university subscription thing.

Restricted Jordan Curve Theorem: A simple closed polygonal curve C consisting of finite number of segments partitions the plane into exactly 2 faces, each have C as boundary.A polygonal curve is a curve composed of finitely many line segments.

 

[ehrenfest]

anyone?