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## IMC 2011 Day 2 Problem 1

Let $(a_n)\subset (\frac{1}{2},1)$. Define the sequence $x_0=0,\displaystyle x_{n+1}=\frac{a_{n+1}+x_n}{1+a_{n+1}x_n}$. Is this sequence convergent? If yes find the limit.

Solution: Prove by induction that $x_n \in (0,1)$ then prove that $(x_n)$ is increasing (substract two neighboring terms and use the recurrence relation). Therefore $(x_n)$ is convergent. Since $(a_n)$ is bounded it has a convergent subsequence. Use this subsequence and the recurrence relation to prove that the only possible limit point for $(x_n)$ is $1$.