I'm starting a class on Advanced Mathematics I next semester and I found a sheet of the class'es 2012 final exams, so I'm slowly trying to solve the exercises in it or find the general layout. I will be posting a lot of questions with the exercises I find challenging, and I would like to ask for any help or methodologies that will make it easier for me to solve.
I understand the whole ordeal is categorized as "homework", but any assistance would be appreciated, as I'm completely clueless and I would like to be prepared.
The following exercise is $Ex. 2$, graded for $8\%$. It's one of the first ones, and I'd guess easier.
Prove that the sequence $(a_n)$ that is defined by the recursive type $a_{n+1} = \dfrac{a_n^2 + 1}{3}\;, a_1 = 1$ converges in $\mathbb{R}$ and find its limit.
I've got no insight on this. No idea where to start or how it's supposed to go. What type of proof would this need?
Let $x=\frac{1}{2}(3-\sqrt{5})$.
Step 1: prove by induction that $a_n$ is decreasing and each $a_n$ is bounded from below by $x$.
Step 2: by Step 1, $\lim_na_n$ exists and equals to some $L$ satisfying $L\geq x$ and $L=\frac{L^2+1}{3}$. You can then verify that $L$ is in fact equal to $x$ above.