${x_n}$ is a sequence such that $|x_{n+1} - x_n| \leq \frac{1}{2}|x_n - x_{n-1}|$. Prove $x_n$ is convergent.

Asked Sep 20 '19 at 10:11

Active Sep 20 '19 at 10:15

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The sequence looks Cauchy. But I do not know how to approach the proof.

edited Sep 20 '19 at 10:15

asked Sep 20 '19 at 10:11

onlinefakir

1

Basically, you have written $1 \leq \dfrac{1}{2}$, which is wrong! – Aniruddha Deshmukh Sep 20 '19 at 10:13
Nothing to prove! – Kavi Rama Murthy Sep 20 '19 at 10:13
$(x_n)$ and $(y_n)$ are convergent sequences. Therefore, $(x_n)$ is convergent. QED – Ennar Sep 20 '19 at 10:14
I am sorry. I mistyped the question – onlinefakir Sep 20 '19 at 10:16
Then for each $p \in \mathbb N*$, what could you say about $\vert x_{n+p} - x_n\vert$ ? – xbh Sep 20 '19 at 10:18
Also: https://math.stackexchange.com/q/2070528/42969 – Martin R Sep 20 '19 at 10:25

0 Answers0