Show that $4 - Un+1 < 1/2(4 - Un)$ Let Un be a sequence such that : U0 = $0$ ; Un+1 = $sqrt(3Un + 4)$ We know (from a previous question) that Un is an increasing sequence and Un < $4$ Show that $4$ - Un+1 <(or =) 1/2(4-Un) I gave it a try and this is what I got : We have that Un is an increasing sequence so : Un > U0 $->$ Un > 0 $->$ $3$Un + $4$ > $4$ $->$ $4 - sqrt(3Un+4)$ < $2$ $->$ $4 - Un+1$ < $2$ And then after that I can't get the right side to be $1/2(4-Un)$ Can someone please show me how I can fix my mistakes and/or how to answer this question?

$$4-\sqrt{3u_n+4}\le \frac 12 (4-u_n)\iff 4-\sqrt{3u_n+4}\le 2-\frac 12u_n\\\ \iff 2-\sqrt{3u_n+4}\le -\frac 12 u_n\iff 2+\frac12 u_n\le \sqrt{3u_n+4} \\\\\iff 4+2u_n+\frac14 u_n^2 \le 3u_n+4 \iff \frac14 u_n^2-u_n\le 0 \\\ \iff u_n\left(\frac14 u_n-1\right)\le0, $$ which holds since $0\le u_n <4.$