Increasing + bounded above ⇒ converges: the star theorem

The theorem

Every increasing sequence bounded above converges (to a finite limit).

Symmetrically: every decreasing sequence bounded below converges.

⚠️ The theorem does not give the limit, only guarantees it exists.

Vocabulary

• $(u_n)$ increasing: $u_{n+1} \geq u_n$ for all $n$.
• $(u_n)$ bounded above by $M$: $u_n \leq M$ for all $n$.
• Convergent: $\lim u_n = \ell$ (finite).

Method

To show $(u_n)$ converges without knowing its limit:

1. Monotonicity: show $u_{n+1} - u_n \geq 0$ (often by induction).
2. Bound: find $M$ with $u_n \leq M$ for all $n$.
3. Conclude by the theorem.

Finding the limit (after proving convergence)

If $u_{n+1} = f(u_n)$ and $u_n \to \ell$, then (if $f$ continuous) $\ell = f(\ell)$.

Solve $\ell = f(\ell)$ to find the limit.

Example

$u_0 = 1$ and $u_{n+1} = \sqrt{u_n + 2}$.

By induction:

• $(u_n)$ is increasing ($u_{n+1} \geq u_n$).
• $(u_n)$ is bounded above by $2$.

So $(u_n)$ converges to some $\ell \leq 2$.

Limit: $\ell = \sqrt{\ell + 2}$, so $\ell^2 = \ell + 2$, thus $\ell^2 - \ell - 2 = 0$, $\ell = 2$ (other root $-1$ ruled out since $u_n > 0$).

Counter-example

$u_n = n$ is increasing but unbounded ⇒ diverges to $+\infty$.

Common mistakes

1. Forgetting to check boundedness (increasing alone isn’t enough).
2. Believing limit = bound (not always).
3. Forgetting continuity of $f$ to identify the limit.

?À toi de jouer

An increasing unbounded sequence:

A.Diverges to +∞B.ConvergesC.May diverge or convergeD.Is eventually constant

Valider