Proving a sequence is increasing Prove the sequence defined by $a_0=1$ and $a_{n+1}=\sqrt{3a_n+4}$ is increasing for all $n\ge0$ and $0\le a_n\le4$
I know that a sequence is increasing if $a_n\le a_{n+1}$ but I don't know what information I can use to prove that since all I have is $a_{n+1}$ and a base case of n=0. Am I able to just test integers 0 to 4 to prove it is true? Or is there another method?

Observe that if $1 < x < 4$, then essentially, $x^2 - 3x + 4 < 0$. Prove by induction that $1 < a_n < 4$. Then, set $x = a_n$ to get that $\{a_n\}$ is increasing.
