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.