MASTER’S ANALYSIS EXAM – JANUARY 2013
3 hours. No notes, textbooks or calculator.
If asked to show something, you must derive it from simpler results. For instance,
you may not prove the intermediate value theorem by quoting a theorem about the
continuous image of a connected metric space.
1. Let Abe the set of algebraic numbers, i.e. real or complex solutions of equations
anxn+· · · +a1x+a0= 0
with integers ajand an6= 0. Show that Ais countable.
2. Let a1, a2, . . . be complex numbers such that
lim
n→∞
n
p|an|<1.
Show that
∞
X
n=1
anconverges absolutely.
3. Let Xand Ybe metric spaces and suppose that f:X→Yis continuous.
Show that if Eis a compact subset of X, then f(E) is compact.
4. Suppose that fis a real continuous function on [a, b]. Prove the intermediate
value theorem: if ylies between f(a) and f(b), there is cin [a, b] such that
f(c) = y.
5. Suppose that the mapping F:A⊂Rn→Rmhas the property that each of
its partial derivatives ∂F
∂xj
exist and are bounded on A. If x∈Ais an interior
point of A, Prove that Fis continuous in x.
6. Using Zx
0
i eitdt, show that
|eix −1|≤|x|(x∈R).
Arguing similarly, obtain
|eix −1−ix| ≤ x2
2(x∈R).
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