Limit of recurrently given sequence

[twoflower]

Hi all,

I don't fully understand solving of limits when the sequence is given by some recurrent expression.

Eg. I have this sequence:

$$a_{n} = \sqrt{2}$$

$$a_{n+1} = \sqrt{2 + a_{n}}$$

$$\lim_{n \rightarrow \infty} a_{n} = ?$$

First, I should prove the monotony and finiteness (is it ok to say it in english this way?). Well I did the proof the monotony by induction, I hope right. Now the finiteness. How should I do it? Can I just guess it won't get greater than. let's say 2 ? Ok, I chose 2 and prove that it is finite.

Now the limit. Our teacher wrote this:

$$\lim_{n \rightarrow \infty} a_{n} = A$$

$$A = \sqrt{2 + A}$$

And I ask, what should this mean? Where does this equality come from?

Of course to solve it is easy and we find out A = 2, which is the limit. But I don't understand why the equality.

Thank you for any help.

 

[e(ho0n3]

If the limit exists then the difference between a[n+1] and a[n] for very, very large values of n is very, very small. In other words, a[n+1] is approximately a[n] for very large n. Hence the substitution. I don't find this very satisfying though. Maybe somebody else can provide a more rigoruous explanation.

 

[danja347]

I think it is like this:
Since $$\lim_{n \rightarrow \infty} a_{n} = A$$ we can say that $$\lim_{n \rightarrow \infty} a_{n+1} = A_{2}$$.
This gives $$A_{2}=\sqrt{(2+A)}$$ and it´s obvious that $$\lim_{n\rightarrow \infty} a_{n} = \lim_{n\rightarrow \infty} a_{n+1}$$ and then we get $$A_{2}=A=\sqrt{2+A}$$ and then we get A=2.

 

[e(ho0n3]

Yes. I think that's it! Very good.