Let n be a natural number.
Prove that 0!+1!+2!+.......n! < (n+1)!
and my < sign should be less than or equal to.
The Attempt at a Solution
The proof is by induction
it works for 0 because 0! is less than or equal to 1!
now we assume it works for n=k by the induction axiom.
now we see if it works for k+1
Now im not sure if I can do this but I will replace
0!+1!+2!+.....k! with (k+1)!
so now I have (k+1)!+(k+1)!<(k+2)!
and k+2 will always be bigger or equal to 2 because k is a natural number.
so the inequality is proved by induction.