How can induction be used to prove that a function is a numerical polynomial?

[Zero266]

Def: A polynomial f(x) with coefficients in Q (the rationals) is called a "numerical polynomial" if for all integers n, f(n) is an integer also.

I have to use induction to prove that for k > 0

that the function f(x) := (1/k!)*x*(x-1)...(x-k+1) is a numerical polynomial

I checked that this is true for k=1, but to be honest I'm not even sure what the dot dot dot means. If k=5 for say, I interpreted the dot dot dot as (1/5!)*x*(x-1)*(x-2)*(x-3)*(x-4). Is this the correct interpretation? If so it is indeed true for k=1, but nonetheless I don't know how to show that it is true for k+1.

Thanks so much.

P.S My professor sucks and it is really discouraging as a recently declared math major. So I will be on here often! Loves.

 

[Mark44]

... is an ellipsis, and means continuing in the same pattern. Your interpretation above is correct.

For an induction proof, you need to show that the statement is true for some base case (n = 1 will do), assume that the statement is true for n = k, and then use that statement to show that the statement is true for n = k + 1. (You'll notice that I changed your k to n.)

The induction hypothesis is that f(x) = 1/k!*x*(x -1)*(x - 2)* ... *(x - k + 1) is a numerical polynomial.

What can you say about 1/(k + 1)!*x*(x -1)*(x - 2)* ... *(x - k)?