2 Countably or Uncountable Infinite? In class, we learned about how infinite sets can have different sizes. Specifically, they can be either countable or uncountable. . One example we saw in class is that the set of all natural numbers is countably infinite. Now prove that the set of all pairs of natural numbers has the same size as the set of natural numbers! . In contrast, an uncountably infinite set is larger than a countably infinite set. Prove that the set of all infinite mathematical series is uncountable infinite. An infinite mathematical series has the form _1 .... For example, the infinite series _ 2 = 1 + 2 + 3 4 ... Another example could be the infinite series _ 1/i = 1/1 + 1/2+ 1/3 + ... The proof must be a proof by contradiction and use diagonalization