Prove that the sequence is increasing and bounded above

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January 15th 2012, 12:32 AM
alexmahone

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.
January 15th 2012, 12:50 AM
girdav

Re: Prove that the sequence is increasing and bounded above

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

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