Basic Analysis I — Real Analysis · Problem 1.2.1

The key insight is that the natural numbers grow without bound, so their reciprocals shrink toward zero. You're really using the Archimedean property—the idea that no matter how small a positive number $t$ is, you can always find a large enough $n$ to make $\frac{1}{n^2}$ even smaller. Start by working backward: what condition on $n$ would guarantee $\frac{1}{n^2} < t$?

The key insight is that the natural numbers grow without bound, so their reciprocals shrink toward zero. You're really using the Archimedean property—the idea that no matter how small a positive number $t$ is, you can always find a large enough $n$ to make $\frac{1}{n^2}$ even smaller. Start by working backward: what condition on $n$ would guarantee $\frac{1}{n^2} < t$?