Uniqueness of the limit

Let $\left(u_n\right)_{n>0}$ be a numerical sequence. Suppose that $\left(u_n\right)$ is convergent^FR. Show that its limit is unique.

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Solution by beignet62FR

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Soient $\ell$ et $\ell'$ deux réels vérifiant la définition de la limite. Supposons $\ell \neq \ell'$ et, quitte à échanger les rôles de $\ell$ et $\ell'$ , supposons que $\ell < \ell'$ .

On prend

$\varepsilon = \frac{\ell' - \ell}{6}.$

Il existe ainsi un rang $n_0$ tel que, pour tout $n \geq n_0$ , on a

$|u_n-\ell| < \varepsilon \qquad\text{et}\qquad |u_n-\ell'| < \varepsilon.$

Or,

$\begin{aligned} |\ell-\ell'| &= |(\ell-u_n)+(u_n-\ell')| \\ &\leq |\ell-u_n|+|u_n-\ell'| \\ &< 2\varepsilon \\ &= \frac{\ell'-\ell}{3}. \end{aligned}$

Ainsi,

$\ell'-\ell < \frac{\ell'-\ell}{3},$

ce qui est absurde puisque $\ell'-\ell>0$ .

Donc nécessairement,

$\boxed{\ell=\ell'}.$

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