Problem # 3
These problems concern approximating arbitrary real numbers by certain kinds of sequences.
a) Prove that for every real number r 2 R there exists a sequence of rational numbers q1 < q2 < : : : < r
with the property qn ! r.
b) Prove that for every real number r 2 R there exists a sequence of irrational numbers z1 < z2 < : : : < r
with the property zn ! r.
Let an > 0 be a sequence of strictly positive numbers. Suppose that = lim an+1
a) Prove that if < 1 then an ! 0.
b) Prove that if > 1 then an ! 1.
c) Give an example with = 1 and an ! 0.
d) Give an example with = 1 and an ! 1.
e) Give an example with = 1 and an bounded but not converging to any limit at all.
Let a1; a2; : : : aN be a collection of real numbers. A weighted average is an expression of the form:
k1a1 + k2a2 + : : : kNaN AN = ;
k1 + k2 + : : : + kN
where all ki 2 N. Note that the case k1 = k2 = : : : = kN = 1 corresponds to the usual notion of average.
a) Prove that if an ! L then the weighted averages above are such that AN ! L regardless of how the
weights ki are chosen.
b) Show that a similar result is true when an ! +1.
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