Apostol calculus volume 1, exercise 1.7 question 1b clarification. How can area of infinite points be zero Question 1 of the exercise asks us to:
“Prove that each of the following sets is measurable and has zero area using the axioms of area in the foregoing section”
Part b of the question is to do so for a set consisting of a finite number of points in a plane.
I have done so by first showing that 1 point has zero area and then using induction to show that a union of all points have zero area.
My question is that although I have proven zero area for the special case of a finite number of points, I also seem to have proven zero area for an infinite number of points by induction which goes against my intuition. Wouldn’t an infinite number of points include say a rectangle of area ten or even just a shape of area infinity? Any clarification would be helpful, thanks.
 A: You've made a common error when first thinking about induction.  You have successfully proved that your statement is true for all finite sets.  That does not mean it's true for all (or even any) infinite sets.  To take the next step, you need some form of transfinite induction.
For example, you can prove fairly easily that all finite sets of natural numbers are bounded.  That proof, even if you do it by induction, will not prove that all infinite sets of natural numbers are bounded.  In fact, as you can easily see, this fails as dramatically as possible -- no infinite set of natural numbers is bounded.
In measure theory you have an extra axiom -- countable additivity -- that helps you take a step toward infinity.  You can sometimes combine countable additivity with a proof about all finite sets to reach conclusions about countable sets.  For example, your proof that all finite sets have measure zero can be easily extended to a proof that all countable sets have measure zero.  Do you see how?  And do you see why this proof can't be extended to tell you anything about uncountable sets?
