Does converge?

I'm having trouble understanding my professor's solution:

exists because extends to a continuous function at x = 0.

Now this is where I'm having trouble. How are these equations derived?

Let be the floor function or greatest integer function. Then

.

Since for , the first term has a finite limit as , by the Alternating Series Test. The second term is bounded in absolute value by

,

so it converges to 0 by the squeeze theorem. So the limit exists, and therefore the integral converges.