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Real Analysis
book

Real Analysis

by V. Karunakaran
May 2024
Intermediate to advanced content levelIntermediate to advanced
585 pages
15h 38m
English
Pearson India
Content preview from Real Analysis
“real: chapter_03” 2011/5/22 22:50 page4—#4
3-4 Real Analysis
Definition 3.2.10 Let {a
n
} be a sequence and S denote the set of all
limit points of this sequence {a
n
} including ±∞. We define the limit
supremum of {a
n
} or the upper limit of {a
n
} (written as lim sup
n→∞
a
n
or
___
lim
n→∞
a
n
) by
lim sup
n→∞
a
n
= sup S
Similarly, the limit infimum of {a
n
}or the lower limit of {a
n
}(written
as lim inf
n→∞
a
n
or lim
___
n→∞
a
n
) is defined as
lim inf
n→∞
a
n
= inf S
Note that for any real sequence {a
n
}, lim sup
n→∞
a
n
and lim inf
n→∞
a
n
always
exist uniquely as elements of [−∞, ∞].
Examples 3.2.11
1. Consider the sequence {a
n
}, where a
n
= (1)
n
. Here
lim sup
n→∞
a
n
= 1 and lim inf
n→∞
a
n
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Publisher Resources

ISBN: 9781299447561Publisher Website