Subsequences and extraction maps

What a subsequence is

A subsequence of $(u_n)$ is obtained by keeping only some of the terms, without ever going backward. Formally, we take a map

$$\varphi : \mathbb{N} \to \mathbb{N} \quad \text{strictly increasing}$$

and consider the sequence $\left(u_{\varphi(n)}\right)_{n \in \mathbb{N}}$. The map $\varphi$ is called an extraction map.

Example: with $\varphi(n) = 2n$, we extract the even-indexed terms $u_0, u_2, u_4, \ldots$; with $\varphi(n) = n^2$, we extract $u_0, u_1, u_4, u_9, \ldots$

The basic lemma

If $\varphi : \mathbb{N} \to \mathbb{N}$ is strictly increasing, then $\varphi(n) \geq n$ for all $n$.

Proof by induction. We have $\varphi(0) \geq 0$ since $\varphi$ takes nonnegative integer values. If $\varphi(n) \geq n$, strict monotonicity gives $\varphi(n+1) > \varphi(n) \geq n$, hence $\varphi(n+1) \geq n + 1$ since these are integers.

This tiny lemma is used everywhere: it is what guarantees that the extraction "moves" toward infinity, so that a subsequence really is a sequence of terms further and further along.

What subsequences preserve

Property of $(u_n)$	True for $\left(u_{\varphi(n)}\right)$?
converges to $\ell$	yes, to the same $\ell$
bounded	yes, by the same bounds
tends to $+\infty$	yes
increasing	yes
diverges	not necessarily

The first row is the most useful, and its contrapositive even more so: if we exhibit two subsequences with different limits, the sequence does not converge.

Canonical example: for $u_n = (-1)^n$, we have $u_{2n} = 1 \to 1$ and $u_{2n+1} = -1 \to -1$. These two limits differ, so $(u_n)$ diverges — without ever having had to work with the definition of convergence.

The even-odd criterion

Conversely, if $\left(u_{2n}\right)$ and $\left(u_{2n+1}\right)$ converge to the same limit $\ell$, then $(u_n)$ converges to $\ell$: every index is even or odd, so the two bounds fit together.

Be careful: the two extractions really must cover all indices. Knowing that $\left(u_{2n}\right)$ and $\left(u_{3n}\right)$ converge to $\ell$ is not enough: index $5$, for example, is reached by neither.

Classic pitfall

Believing that an extraction map must be surjective, that is, must reach every integer. This is false: $\varphi(n) = n^2$ skips index $2$, and yet $\left(u_{n^2}\right)$ is a perfectly legitimate subsequence. The only constraint is strict monotonicity — and it is what must be checked carefully in the construction of the next chapter.