Prove sequence defined by recurrence relation using induction

Question

Confused at this question, from what I gather strong induction is necessary here to prove this but the algebraic step after the Inductive Hypothesis is where I'm not too sure.

Basis:
2 <= a₁ = 2 <= 4
2 <= a₂ = 2 <= 4

Inductive Hypothesis:
Assume that for any 2 < n <= k, we have that 2 <= a_n <= 4

Show a_n+1 is true.

Answer 1

Your base case is correct.

Now, fix an $n\geq 3$ and suppose that for all $k<n$, $2\leq a_{k}\leq 4$. We want to show that $2\leq a_{n}\leq 4$. Note that $$ a_{n}=\frac{1}{2}\left(a_{n-1}+\frac{8}{a_{n-2}}\right)\leq\frac{1}{2}\left(4+\frac{8}{2}\right)=4 $$

and

$$ a_{n}=\frac{1}{2}\left(a_{n-1}+\frac{8}{a_{n-2}}\right)\geq \frac{1}{2}\left(2+\frac{8}{4}\right)=2, $$ by the inductive hypothesis.

Hence, $2\leq a_{n}\leq 4$, as desired.