Convex optimization book - solution - exercise - 2.3 - midpoint convexity

The following video is a solution for exercise 2.3 from the seminal book “convex optimization’’ by Stephen Boyd and Lieven Vandenberghe”. This problem is classified under “definition of convexity” in the exercises section of chapter 2.
What does the problem say? This problem is tries to add a mild condition (closedness) to a set C which is midpoint convex and make that set to be convex.
What is the approach? We show that convexity implies midpoint convexity but the other way around is not true. To show midpoint convexity does not imply convexity I will mention a counterexample. Then I will sketch the proof pictorially to make it more understandable.

[darrirro]

8:20 z2 = x/2 + x/4 + y/4 btw no y/2, but thanks for the explanation <3

[hank-pmrd]

07:51 z1 is the midpoint of x and y, why do you say "if z1 is closer to x or closer to y"

[孙天成-gz]

May I ask why Lamda k for k to the infinity then it becomes Lamda? is limZk for k to the infinite means that to cut the line for the infinite times so that we can get any point on the line?