# Thread: Prove that the sequence is increasing and bounded above

1. ## Prove that the sequence is increasing and bounded above

Define a sequence by

$a_{n+1}=\frac{a_n+1}{2}$, $n\ge 0$

Prove that if $a_0\le 1$, the sequence is increasing and bounded above.

My solution:

$a_{n+1}-1=\frac{a_n+1}{2}-1=\frac{a_n-1}{2}$

So, $a_0\le 1\implies a_n\le 1$ for all $n$, ie the sequence is bounded above by 1.

$a_{n+1}-a_n=\frac{a_n+1}{2}-a_n$

$=\frac{1-a_n}{2}$

We know that $a_n\le 1$ for all $n$. So, $a_{n+1}\ge a_n$ and the sequence is increasing.

2. ## Re: Prove that the sequence is increasing and bounded above

Show by induction that $a_n\leq 1$ for all $n$.