Let $b_1=1$ and $b_n=1+\frac{1}{1+b_{n-1}}$
Prove that $(b_{2k-1})$ is increasing for $k \in \mathbb{N}$
By definition, a sequence $(a_{n})$ is increasing if $a_{n}≤a_{n+1}$ for all $n \in \mathbb{N}$.
SO, for this problem, must prove $b_{2n-1}≤b_{2n}$ for all $n$.
Proceed by induction:
Start with $n=1$. Then, $b_1=1$ and $b_2=3/2$, so $b_1≤b_2$.
Assume inductively that $b_{2n-1}≤b_{2n}$, prove $b_{2n}≤b_{2n+1}$.
Am I doing this correctly? I want to know before I continue.
Thanks.