You may want to look at showing its true for n = 2 instead since you are looking at k-1.
Also for fib(k-1), use the fact that this is less than (k-1)! and group the terms to show that k! + (k-1)! < (k+1)! You've nearly completed the proof, but you've left out this critical step which will show that is true for all later values of k since fib(k) <= k! and fib(k-1) <= (k-1)!.
So for the first step look at when n = 2 not n = 0 so basically show 0+1+2 < 2! since you are dealing with k+1 (which is n), k, and k-1.
Otherwise though you have the right idea.