Prove that the sequence$ c_1 = 1$, $c_{n+1} = 4/(1 + 5c_n) $ , $ n \geq 1$ is convergent and find its limit Prove that the sequence $c_{1} = 1$, $c_{(n+1)}= 4/(1 + 5c_{n})$     , $n \geq 1$  is convergent and find its limit.
Ok so up to now I've worked out a couple of things.
$c_1 = 1$
$c_2 = 2/3$
$c_3 = 12/13$
$c_4 = 52/73$  
So the odd $c_n$ are decreasing and the even $c_n$ are increasing.  Intuitively, it's clear the the two sequences for odd and even $c_n$ are decreasing/increasing less and less.
Therefore it seems like the sequence may converge to some limit $L$.
If the sequence has a limit, let $L=\underset{n\rightarrow \infty }{\lim }a_{n}.$   Then $L = 1/(1+5L).$
So we yield $L = 4/5$ and $L = -1$.  But since the even sequence is increasing and >0, then $L$ must be $4/5$.
Ok, here I am stuck.  I'm not sure how to go ahead and show that the sequence converges to this limit (I tried using the definition of the limit but I didn't manage) and and not sure about the separate sequences how I would go about showing their limits.
A few notes :
I am in 2nd year calculus.
This is a bonus question, but I enjoy the challenge and would love the extra marks.
Note : Once again I apologize I don't know how to use the HTML code to make it nice.

So $c_{1} = 1$ and $c_{n+1} = 4/(1+5c_{n})$ for $n \geq 1$. Let $C(x) = \sum_{n \geq 1} c_{n}x^n$. Then maybe try to express $\sum_{n \geq 1} c_{n+1}x^n$ and $\sum_{n \geq 1} \frac{4x^n}{1+5c_{n}}$ in terms of $C(x)$ to get a closed form. 
