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# Prove an inequality for the absolute value of derivatives of a function

Let and assume is a function with continuous in such that

Furthermore, assume is maximal at a point (i.e., does not have its maximum at either endpoint of the interval). Prove that

Proof. Since attains its maximum on the interval we know there is some such that . Then,

Evaluating these integrals separately, we have (by the first fundamental theorem of calculus, which is permissible since is continuous by hypothesis)

Now, we use the bound for all ,

Next, we evaluate the second integral,

And so,

Therefore,

# Prove that every fixed real number is between two integers.

Prove that if is fixed, then there exist such that .

Proof. Since the set of positive integers is unbounded above (Example #1, p. 24 of Apostol) we know there exists an such that (otherwise would be an upper bound on ).
Then, by the same logic there is some such that ; hence, . Since , we know . Thus, we have found such that

# The reals are unbounded above.

Prove that there is no such that for all .

Proof.
Let be given. Then, since (Theorem I.21) we have for all (Theorem I.18). Then take and we have ; hence . Hence, there is no such that for all