MASTER’S EXAM, ANALYSIS, JANUARY 2011
1. When we say that (R,+,·,≤) is a complete ordered field, we are asserting more
than that (R,+,·) is a field and (R,≤) is a totally ordered set. What else are
we asserting? (Be specific.)
2. Prove that if {an}∞
n=1 is a sequence of real numbers converging to Aand {bn}∞
n=1
is a sequence of nonzero real numbers converging to B6= 0, then {an/bn}∞
n=1
converges to A/B.
3. Let f:X→Ybe a continuous function between metric spaces. If Xis
compact, prove that fis uniformly continuous.
4. Let f:Rn→Rnbe a continuous function satisfying kf(x)k<kxkfor every
x6=0. For some initial point x0∈Rndefine the sequence {xk}∞
k=0 recursively
by the rule xk+1 =f(xk). Show that the sequence converges to 0.
5. Let f:R→Rbe a uniformly continuous function on R. Prove that there exists
positive constants aand bsuch that
|f(x)| ≤ a|x|+b, ∀x∈R.
6. Let U⊂R2be a nonempty open set. Prove that there is no one-to-one
continuously-differentiable function mapping Uinto R.
7. Let f: [0,1] →Rbe defined by the formula
f(x) = (0 if x /∈Q
1/q if x∈Qand x=p/q in lowest terms.
Determine with proof the value of the (Riemann-Darboux) integral R1
0f(x)dx
or that the integral is undefined.
8. Evaluate Z1
0Z1
√yZ1
x3
√x3+z dz dx dy.
9. For a > b > 0, evaluate the integral Z∞
−∞
cos x dx
(x2+a2)(x2+b2).
10. Suppose that fis analytic in |z|<1 and that |f(z)|<1
1− |z|for |z|<1. Show
that for 0 < R < 1,
|f(n)(0)| ≤ 1
Rn(1 −R)(n= 1,2, . . .).
What choice of Ryields the best upper bound?