What is Cauchy sequence?

Example 11.19 (Verifying the Cauchy property by hand)

Let $u_n = \sum_{k=0}^{n} \frac{\cos k}{2^k}$ — no monotonicity, no guessable limit. For $p > q$:

by the triangle inequality, $\abs{\cos k} \leq 1$ and a finite geometric sum. Given $\varepsilon > 0$, choose $N$ with $2^{-N} \leq \varepsilon$: all gaps beyond $N$ are $\leq \varepsilon$, the sequence is Cauchy, hence converges — to a limit nobody can name in closed form, which is exactly the point. The closing insight: geometric domination of the increments is the standard way to earn the Cauchy property, and Chapter 17 will bottle the argument as “absolute convergence implies convergence”.

Example 11.22 (A Cauchy sequence with an invisible limit)

Let $S_n = \sum_{k=1}^{n} \frac{1}{k^2}$. For $p > q \geq 1$:

so beyond $N > \frac1\varepsilon$ all gaps are $\leq \varepsilon$: $(S_n)$ is Cauchy, hence converges. Notice what just happened: we proved that a specific real number exists without having any name for it. (It is $\frac{\pi^2}{6}$ — a celebrated identity of Euler, proved in the Year 2 volume; nothing in this chapter could tell us that.) This division of labor — existence now, identification later, if ever — is the Cauchy criterion’s whole point, and the engine of the theory of series in Chapter 17.