Probing Induction (25 Points) Recall the Fibonacci numbers from class, defined by f 0
​
=1,f 1
​
=1, and f n+2
​
=f n+1
​
+f n
​
for n≥0. In class we showed that 2
1
​
(1.5) n
≤f n
​
≤2 n
for all n≥0. This gives a nice bound on how fast the Fibonacci numbers grow. 1) Find α,β as small as possible so that you can prove f n
​
≤α∗β n
for all n≥0. (Focus on choosing β as small as possible, then choose α to work with that β). 2) Find γ,δ as large as possible so that you can prove γ∗δ n
≤f n
​
for all n≥0. (Focus on choosing δ as large as possible, then choose γ to work with that δ.) 3) What is the limit of n
1
​
lnf n
​
as n→[infinity] ?