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putnam120 · #7 September 18th, 2007, 05:54 AM

First we show that the sequence is bounded.

$\left|\frac{a_n}{a_{n-1}}\right|\Longrightarrow 1<\left|1+\frac{a_{n-2}}{a_{n-1}}\right|<2$ since each term in the sequence is obviously larger than the previous term.

now define $R_n=\frac{a_n}{a_{n-1}}$ and look at the ratio $\frac{R_{n+1}}{R_n}=\frac{a_{n+1}a_{n-1}}{a_n^2}$ which is just $\frac{a_n^2-1}{a_n^2}<1$ . (easy induction proof ill leave the verification to the reader, but if this were for some competition i would include the proof

) so the sequence is monotonically decreasing.

and every bounded monotonic sequence converges.