**2 Exercise** *Suppose is infinitely differentiable. Suppose, furthermore, that for every , there is such that . Then is a polynomial.* (Hint: Baire's category theorem.)

**Exercise** * and are irrational numbers. Moreover, is neither an algebraic number nor p-adic number, yet is a p-adic number for all p except for 2.*

**Exercise** *There exists a nonempty perfect subset of that contains no rational numbers. (Hint: Use the proof that e is irrational.)*

**Exercise** *Construct a sequence of positive numbers such that converges, yet does not exist.*

**Exercise** *Let be a sequence of positive numbers. If , then converges.*

**Exercise** *Prove that a convex function is continuous (Recall that a function is a* convex function *if for all and all with , )*

**Exercise** *Prove that every continuous function* f *which maps [0,1] into itself has at least one fixed point, that is such that *

Proof: Let . Then

**Exercise** *Prove that the space of continuous functions on an interval has the cardinality of *

**Exercise** *Let be a monotone function, i.e. . Prove that has countably many points of discontinuity.*

**Exercise** *Suppose is defined on the set of positive real numbers and has the property: . Then is unique and is a logarithm.*