#1
The goal of this problem is to ensure mastery of the basic concepts concerning sequences. Recall
Definition:
A sequence
converges to a real number
L
if for every , there is some
positive integer
N
so that for all
, .
Exercise:
Verify, using the
definition of limit of a sequence, that
a.
b.
c.
Exercise:
If
converges to
A
, prove that
converges to . Show by example that the converse need not be true.
Exercise:
Give an example for each of the following or prove that no such example exists:
a.
A bounded sequence that does not converge
b.
A bounded sequence that does converge
c.
A sequence that converges to a real number
L
but is not bounded
d.
A monotone sequence that is Cauchy
e.
Sequences
and
which do not converge, but
converges
f.
Repeat part
e
. replacing
with
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#2
The goal of this problem is to ensure mastery of the basic concepts concerning limits of functions.
Recall
Definition:
Let
be some function and let
p
be an accumulation point of
D
. Then
f
has
limit
L
at
p
if for every , there is
such that for all
satisfying , it follows that .
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 Spring '08
 PLOTKIN
 Topology

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