prove lim_(n→∞) x^n = 0 when ∣x∣


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Question Number 197808 by mokys last updated on 29/Sep/23

$p r o v e \underset{n \to \infty}{l i m} x^{n} = 0 w h e n ∣ x ∣ < 1$
Commented by Noorzai last updated on 30/Sep/23

$p l e a s e g i v e m o r e i n f o r m a t i o n$
	Answered by mr W last updated on 30/Sep/23

	$s a y ∣ x ∣= a w i t h 0 < a < 1$ $1 > a > a^{2} > . . . a^{n} > a^{n + 1} > . . . > 0$ $\Rightarrow \lim_{n \to \infty} a^{n} = 0$ $- a^{n} ⩽ x^{n} ⩽ a^{n}$ $\lim_{n \to \infty} (- a^{n}) ⩽ \lim_{n \to \infty} x^{n} ⩽ \lim_{n \to \infty} a^{n}$ $0 ⩽ \lim_{n \to \infty} x^{n} ⩽ 0$ $\Rightarrow \lim_{n \to \infty} x^{n} = 0$
	Commented by mokys last updated on 30/Sep/23

	$t h a n k y o u$
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