Intereting Posts

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I am trying to prove it by induction, but I’m stuck

$$\mathrm{fib}(0) = 0 < 0! = 1;$$

$$\mathrm{fib}(1) = 1 = 1! = 1;$$

**Base case** n = 2,

$$\mathrm{fib}(2) = 1 < 2! = 2;$$

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**Inductive case** assume that it is true for ~~(k+1)~~ $k$

Try to prove that $\mathrm{fib}(k+1) \leq(k+1)!$

$$\mathrm{fib}(k+1) = \mathrm{fib}(k) + \mathrm{fib}(k-1) \qquad(LHS)$$

$$(k+1)! = (k+1) \times k \times (k-1) \times \cdots \times 1 = (k+1) \times k! \qquad(RHS)$$

……

How to prove it?

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$$

F_{k+1} = F_k + F_{k-1} \le k! + (k – 1)! \le k! + k! \le 2 k! \le (k + 1) k!

$$

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