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Squeeze rule for sequences
Let
be three sequences of real numbers such that
for all . If
and
exist and are equal, say to , then
also exists and equals .
The proof is fairly straightforward. Let be any real number . By hypothesis there exist
such that
 for all 
 for all 
Write
. For we have
- if
:
- else
and:
So, for all , we have
, which is the desired conclusion.
Squeeze rule for functions
Let
be three real-valued functions on a neighbourhood of a real number , such that
for all
. If
and
exist and are equal, say to , then
also exists and equals .
Again let be an arbitrary positive real number. Find positive reals and such that
 whenever 
 whenever 
Write
. Now, for any such that
, we have
- if
:
- else
and:
and we are done.
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"squeeze rule" is owned by Daume. [ owner history (1) ]
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(view preamble)
| Other names: |
squeeze theorem, squeeze test |
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Cross-references: positive, neighbourhood, functions, conclusion, hypothesis, real numbers, sequences
There are 3 references to this entry.
This is version 1 of squeeze rule, born on 2003-07-20.
Object id is 4483, canonical name is SqueezeRule.
Accessed 6599 times total.
Classification:
| AMS MSC: | 26A03 (Real functions :: Functions of one variable :: Foundations: limits and generalizations, elementary topology of the line) |
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Pending Errata and Addenda
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