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Math 311:
Review Problems for Final
#1
Let
S
be a nonempty subset of real numbers that is bounded above. Put . Prove that for every , there is
some
x
in
S
satisfying .
#2
Let
A
and
B
be nonempty subsets of real numbers that are bounded above. Then it is appropriate to put
and . Prove that
where . Is it true that ? Justify your answer.
#3
Let
S
be a nonempty subset of real numbers that is bounded above. Put
a.
Show that where .
b.
Show that
where
#4
Give an
proof that
converges to .
#5
State and prove the Squeeze Theorem for sequences
#6
State and prove the Nested Interval Theorem
#7
State and prove the Monotone Convergence Theorem
#8
Let
x
be any positive real number and define a sequence
by
where
represent the largest integer less than or equal to
x
, i.e. . Prove that
converges to
x
/2.
#9
(Cesaro Means)
Show that if
is a convergent sequence, then the sequence
defined by the arithmetic
means
must converge to the same limit. Give an example where the sequence
diverges but the sequence
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 Spring '08
 PLOTKIN
 Real Numbers, Sets

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