# Proof by induction.

## Homework Statement

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
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.