Hermite's identity

In mathematics, Hermite's identity, named after Charles Hermite, gives the value of a summation involving the floor function.

It states that for every real number x and for every positive integer n the following identity holds:[1][2] Split

into its integer part and fractional part,

with By subtracting the same integer

from inside the floor operations on the left and right sides of this inequality, it may be rewritten as Therefore, and multiplying both sides by

gives Now if the summation from Hermite's identity is split into two parts at index

, it becomes Consider the function Then the identity is clearly equivalent to the statement

But then we find, Where in the last equality we use the fact that

It then suffices to prove that

But in this case, the integral part of each summand in

is equal to 0.

We deduce that the function is indeed 0 for all real inputs