Lesson 49 — Limit of Sequences (Formalized)

Sequences, convergence, and theorems

Key theorems

Cauchy

In $\mathbb{R}$: Cauchy $\iff$ convergent (completeness). In $\mathbb{Q}$: not every Cauchy converges ($a_n$ approximating $\sqrt 2$).

Monotonicity + boundedness

• $(a_n)$ increasing and bounded above $\Rightarrow$ converges to $\sup a_n$.
• $(a_n)$ decreasing and bounded below $\Rightarrow$ converges to $\inf a_n$.

Subsequences

$(a_{n_k})$ is a subsequence of $(a_n)$ if $n_1 < n_2 < \ldots$. $\lim a_n = L \Rightarrow \lim a_{n_k} = L$ for every subsequence.

Divergent sequences

• $a_n = (-1)^n$: oscillates between $-1$ and $1$. Subsequences converge (to $\pm 1$), but $a_n$ does not.
• $a_n = n$: grows without bound ($\to +\infty$).
• $a_n = n \sin(n)$: irregular behavior, no limit.

Recursive sequences

$a_{n+1} = f(a_n)$, $a_0$ given. The limit, if it exists, is a fixed point of $f$: $L = f(L)$.