## SEEMOUS 2011 Problem 4

Let be a twice continuously differentiable increasing function. Define the sequences given by

and

for .

The interval is divided into three equal segments. Prove that for large enough , the number belongs to the middle one of these segments.

*SEEMOUS 2011 Problem 4*

**Solution:** Define . Then we can write . We apply Taylor’s expansion formula of order 2 for in points .

We have . For we get . Summing this for we get , where is a Riemann sum for .

The middle intervals are . If the function is constant, and we know that is increasing and therefore, non-constant (increasing means strictly increasing). Therefore .

Now, we calculate , and taking the limit for we get that , which means that for great enough we have . On the other hand, .

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