Dear friends, I have a problem with the following question.

I will denote

and call by -th convergent.

And define

and

It is easy to show that

is true.

We will apply induction.

Clearly, the claim holds for .

Suppose that

holds for .

Then, we have

Thus, the claim is true.

Now my question comes.

Let be an irrational number, and set

where is the least integer function.

How to show that holds, where and are defined as above with .