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I attempted using induction to prove it formally but was not sure how to proceed.

Additionally, how can you prove the sequence is neither monotonically increasing nor decreasing?

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closed as off-topic by Martin R, Aqua, Claude Leibovici, Jack D'Aurizio Oct 18 '17 at 8:39

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After checking of the base and after assuming of the induction we obtain the following.

For all $n\geq3$ we have $$a_n=\frac{a_{n-1}+a_{n-2}}{2}\geq\frac{1+1}{2}=1.$$ Also, for all $n\geq3$ we have: $$a_n=\frac{a_{n-1}+a_{n-2}}{2}\leq\frac{2+2}{2}=2.$$ Done!

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  • $\begingroup$ To argue this formally you would have to use induction, but yeah this is a general idea. $\endgroup$ – Siddhartha Oct 18 '17 at 8:37

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