Countability of a Set of Square Reciprocals

Show that the set {$\frac{1}{1^2}$, $\frac{1}{2^2}$, $\frac{1}{3^2}$, ...} is countable by creating a bijection to the natural numbers.

To show that a set is countable, we typically want to establish a bijection between the set in question and the natural numbers. In this problem, the set of square reciprocals is considered. A bijection is a one-to-one and onto mapping that effectively 'counts' each element of the set using the natural numbers. Understanding countability is crucial in real analysis as it distinguishes between different sizes of infinity, like the countable infinity of natural numbers versus the uncountable infinity of real numbers. In approaching this problem, you'll want to define a function that takes a natural number and maps it to an element of the set in question. Ensuring that this function is both injective (one-to-one) and surjective (onto) will confirm the countability of the set. It's also essential to understand the implications of determining a set as countable in terms of its cardinality, as this contributes to the broader understanding of set theory and its application in real analysis.