Lebl-Basic-Analysis-I · Problem 1.2.1

The core idea here is the Archimedean property of the real numbers: no matter how small a positive number is, you can always find a natural number large enough to make its reciprocal smaller still. Since $\frac{1}{n^2}$ shrinks as $n$ grows, you just need to find an $n$ where $n^2$ is large enough that $\frac{1}{n^2}$ dips below your given threshold $t$. This is really about showing that the natural numbers are unbounded.

The core idea here is the Archimedean property of the real numbers: no matter how small a positive number is, you can always find a natural number large enough to make its reciprocal smaller still. Since $\frac{1}{n^2}$ shrinks as $n$ grows, you just need to find an $n$ where $n^2$ is large enough that $\frac{1}{n^2}$ dips below your given threshold $t$. This is really about showing that the natural numbers are unbounded.