Thank you for interesting proof by using interesting theorem. I have found two following proofs:

1. y = x^5 - x = (x - 1)*x*(x + 1)*(x^2 + 1)
Therefore y = 0 (mod 5) and y = 0 (mod 2). It means x^5 and x are ended with the same digit.

2. (x + 1)^5 = (x^5 + 5*x^4 + 10*x^3 + 10*x^2 + 5*x + 1) =
5*x*(1 + x)*(1 + x + x^2) + (x^5 + 1)
Easy to show 5*x*(1 + x)*(1 + x + x^2) = 0 (mod 10)
Therefore the last digit of (x + 1)^5 is the last digit of x^5 plus 1. Start with x = 0 we can get all the last digits of 0, 1, 2, ... 9 are the same 0, 1, 2,...9

May be exist also another interesting proofs?

I am thinking now about two small questions:
1. Why after one, five is the first digit with this property?
2. Why other one, only 5 and 6 hold the property: the last digit of x^n is the same x with any n?

Best regards,
Bui Quang Tuan