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Problem Math Set 0
9/4/2007
Solutions are due at the beginning of class on Friday, September 7, 2007. 1. In this problem, A is a non-empty subset of R. A real-valued function g defined on A is bounded if g = sup |g(x)| < . Also in this problem, {fn }n1 is a sequence of
xA
bounded real-valued functions defined on A with bn = fn
.
a) Suppose that {fn } converges uniformly on A to a function, f . Prove that f is bounded, and, if b = f , b = lim bn .
n
b) Suppose that {fn } converges pointwise on A to a function, f . Must f be bounded? Either prove that f is bounded, or give a counterexample. c) Suppose that {fn } converges pointwise on A to a bounded function, f , and that b = f . Must b = lim bn ? Either prove that this equality is correct, or give a n counterexample. 2. Part of an open cone with vertex the origin and aperture or opening is shown in the diagram to the Give right. a precise definition (in either R2 or C) of such an object (it is open, so only the inside of the cone should be included!). Fix in (0, ), and suppose W is an open cone with aperture . Prove the following reverse triangle inequality (using algebraic manipulation in either R2 or C): There is C > 0 depending on (and not on W ) so that if z1 and z2 are in W , then C (|z1 | + |z2 |) |z1 + z2 |.
3. Suppose {tn }n1 is a sequence of positive real numbers. Prove that there is a power series
n=0 x is greater than tn whe...