1. The problem statement, all variables and given/known data Let n be a natural number. Prove that 0!+1!+2!+.......n! < (n+1)! and my < sign should be less than or equal to. 3. 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 0!+1!+2!+.....k!+(k+1)!<(k+1+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)! 2(k+1)!<(k+2)! 2(k+1)!<(k+2)(k+1)! and k+2 will always be bigger or equal to 2 because k is a natural number. so the inequality is proved by induction.