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?

Just a quick sketch, not a complete solution.   
Try an induction on $n$ for showing that $0\le a_n\le4$.
This is easy.   
Once done, write that $a_{n+1} > a_n$ and try to prove it.
After some trivial rework, it's equivalent to $(a_n+1)(a_n-4) < 0$
which is true (from the induction which you completed).   
