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(As we have defined earlier, if it is bounded above and below,
then x
n
is called a bounded sequence.)
Theorem 1 (Monotonic Sequence Theorem)
Every bounded, monotonic sequence has a limit.
To be more specific, if x
n
increases and is bounded from above
then it converges. If it decreases and is bounded from below then it
converges.
Example 3
Discuss how both of the above statements follow from the just
stated MS Theorem.
Theorem 2 (Squeeze Theorem for Sequences)
If a
n
b
n
c
n
for and n n
0
and lim
n→∞
a
n
= lim
n→∞
c
n
= L then
lim
n→∞
b
n
= L.
Another useful fact about limits of sequences is given by the
following theorem, which follows from the Squeeze Theorem because
−|a
n
| a
n
|a
n
|.
Theorem 3
If lim
n→∞
|a
n
| =0,then lim
n→∞
a
n
=0.
Example 4
Let x
n
=
1+
1
n
n
. The base tends to1 as n →∞, the exponent
goes to ∞. We are thus in a 1
∞
indeterminate situation. To convince
yourself, just apply the natural logarithm:
y
n
=lnx
n
= n ln
1+
1
n
to get an indeterminate situation 0 ·∞ (whichwehavebeenalready
dealing with).
7
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