<s>[INST] To prove $x_n<3$ for sequence $x_{n+1} = \frac{12(1+x_n)}{13+x_n}$ by induction Prove $x_n<3$ for a sequence  given by $$x_{n+1} = \frac{12(1+x_{n})}{13+x_{n}}$$ where $x_1$ is positive real number less than $3$.
For $n = 1$ statement is trivial, but I am stuck at doing further.  Thanks   
 [/INST] $$\frac{12(1+x_n)}{13+x_n}=\frac{12(13+x_n)}{13+x_n}-\frac{12\cdot 12}{13+x_n} = 12 - \frac{12\cdot 13}{13+x_n} < 12 - \frac{12\cdot 12}{16} = 12-9=3$$
To see the inequality, notice that $x\mapsto \frac1{13+x}$ is decreasing, $x\mapsto -\frac1{13+x}$ is increasing. (For $x>-13$.)
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