A proof of an area as a set function

[Shing]

Hi guys, recently, I am a freshmen majoring in Physics, recently using Apostle Calculus to self-study. However, I am having a hard time knowing whether if my proof is correct or not, since there isn't any solution.

Homework Statement

Prove that following set is measurable and has zero area: a set consisting of a single point.

The Attempt at a Solution

First, I have considered the Axiom of Choice of scale{Every rectangle R is in Measurable Set. If the edges of R have lengths h and k then a(R)=hk} to show that h and k both is zero therefore, a(Point)=0

but I couldn't convince myself that a point is an rectangle! So my former approach is not proper.

and I couldn't think up any other approaches so far.

would anyone be king enough to give me a few suggestions?

many thanks!

 

[Office_Shredder]

Think about making really small rectangles around the point

 

[Shing]

Think about making really small rectangles around the point

Thank you very much indeed!

so here is my new approach, what would the problem be in my proof? :
let there be a point in a rectangle $a^2$
$$\exists a\in R$$
s.t. for all $x\in R, 0<a<x$
$$\implies a^2=0$$
$$\implies f(point)=0$$