# Convergent Sequence Question

1. Feb 12, 2013

### gajohnson

1. The problem statement, all variables and given/known data

Let $S_{1}=1$ and $S_{n+1}=\sqrt{2+S_n}$

Show that $\left\{S_n\right\}$ converges and find its limit.

Hint: First assume that the limit exists, then what is the possible value of the limit? Second, show that the sequence is increasing and bounded. Finally, follow the definition of convergence to show that the sequence converges.

2. Relevant equations

NA

3. The attempt at a solution

Well it is pretty clear that this converges to 2, so that's a start.

I am having difficulty constructing a good way to show that the sequence is increasing and bounded. Any help getting started would be nice.

Thanks!

2. Feb 12, 2013

### Dick

To show it's increasing you want to show sqrt(x+2)>x, right? For what range of x is that true? Try to solve the inequality.

3. Feb 12, 2013

### gajohnson

Well because $S_1=1$ is given, the sequence is strictly increasing for $x\in[1,2)$, and the sequence is monotonically increasing for $x\in[1,2]$.

Is showing this by solving the inequality enough to claim that the sequence is increasing and also bounded by 2 (since solving the above as an equality gives 2)?

4. Feb 12, 2013

### Dick

Yes, showing the inequality for the range x in [1,2) will show it. To show it's bounded you need to show the inequality sqrt(x+2)<2 holds in that range.

5. Feb 12, 2013

### gajohnson

I believe I've got it now. Thanks for your help!