<s>[INST] How can I prove this sequence convergent? I am trying to prove the sequence $(u_n)$ so that $u_1=1$ and $u_{n+1}=1+\dfrac{1}{u_n}$ is convergent. With some first items, I guess that the subsequence $(u_{2n})$ is decreasing and the subsequence $(u_{2n-1})$ is increasing, but I can't proof this. How can I proof?
 [/INST] Hint: We can write
$$u_{3} = 1 + \frac{1}{1 + \frac{1}{u_1}} = 1 + \frac{u_1}{u_1+1} = 
\frac{u_1+1}{u_1+1} + \frac{u_1}{u_1+1}
= \frac{2u_1+1}{u_1+1}$$
$$u_{4} = 1+\frac{u_1+1}{2u_1+1} = 
\frac{2u_1+1}{2u_1+1} + \frac{u_1+1}{2u_1+1} =
\frac{3u_1+2}{2u_1+1}$$
$$u_{5} = 1 + \frac{2u_1+1}{3u_1+2} =
\frac{3u_1+2}{3u_1+2} + \frac{2u_1+1}{3u_1+2} = 
 \frac{5u_1+3}{3u_1+2}$$
Notice a pattern? You can prove it using induction!
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