Given the sequence: a(1) = 1, a(n+1)=0,5(a(n)+2/a(n))
I have found through speculations that the limit value is SQRT(2).
The Attempt at a Solution
I started by proving that for n>1; a(n+1) < a(n) and also proved that for n>1 the sequence is limited from below, by SQRT(2).
Now I have the conditions for the rule: if a function(or sequence(i think/hope)) f is decreasing, and is limited from below, it has the limit value;
Lim(f) = A when x-->infinity where A = inf(f(x)) where x is close to infinity.
Now the problem, I had two possible routes:
1: Find inf(f), I started by proving that for all n>1, a(n)>=SQRT(2)
then I made a K>SQRT(2) and tried proving that there existed an n+1>1 where a(n)<K, I am not sure of that but I think it can be done by choosing a domain for K and solving it for K, I got for a(n+1):
K-SQRT(k^2-2) < a(n) < K+SQRT(k^2-2) and this is where I was at so far.
2: Find the limit value of a(n) with conventional methods, whatever those are, because I have apparantly done so a year ago when we were yet to explore sup and inf of functions, somehow you are suppost to get;
Lim(f) = SQRT(2) when x-->infinity but I'm not sure how you do that.
If anybody has a good tutorial or explaination on this, or help in general it would be greatly appreciated, I am suppost to give in my maths report tomorrow morning,