Recursive sequence. Need help finding limit. This is my recursive sequence:
$a_1=\frac{1}{4};\space a_{n+1}=a_n^2+\frac{1}{4}$
for $n\ge 1$
In order to check if this converges I think I have to show that
1) The sequence is monotone increasing/decreasing
2) The sequences is bounded by some value
For 1) I am going to use the ratio test.
$\frac{a_{n+2}}{a_{n+1}}>1$ $\implies$ monotone increasing
$\frac{a_{n+2}}{a_{n+1}}<1$ $\implies$ monotone decreasing
$\frac{(a_{n+1})^2+\frac{1}{4}}{a_{n+1}}=a_{n+1}+\frac{1}{4}>0$ $\implies$monotone increasing
I am really not sure about this. How would I checkt/show it is bounded by some value?

$a_{n+1}-a_n=\frac{(2a_n-1)^2}{4}> 0$. So this is a monotone increasing sequence. Now to see whether the sequence is bounded or not, observe that the limiting value should satisfy $a=a^2+1/4\implies a=1/2$. So, let the sequence be unbounded. Then $\exists N$ such that $a_{N-1}\le 1/2,\ a_N>1/2$. But $a_{N}>1/2\implies a_{N-1}>1/2$ which leads to a contradiction. Hence the sequence is bounded and converges to $1/2$.
