What is Subsequence?

Example 11.15 (Subsequential limits)

For $u_n = (-1)^n \frac{n}{n+1}$: the even subsequence tends to $1$, the odd one to $-1$, so the sequence diverges — but it does so in an organized way, clustering around the two values $\pm 1$. For $u_n = \cos\frac{2\pi n}{3}$: the three subsequences of indices $3k$, $3k + 1$, $3k + 2$ are constant, equal to $1$, $-\frac12$, $-\frac12$; the set of subsequential limits is $\{1, -\frac12\}$. The closing insight: a bounded sequence converges exactly when it has a single subsequential limit (Exercise 11.8); divergence of a bounded sequence always means at least two clusters, and Bolzano–Weierstrass below guarantees there is at least one.