Math
Alternating Series Test
Archimedean Property
Bolzano Weismann Theorem
Cauchy Sequences
Fundamental Trigonometric Limit
Limits at Infinity and Asymtotes
Limits of a Function
Limits of a Sequence
Limits of Rational Functions
LUB GLB
Monotone Convergence Theorem (MCT)
Monotonic Sequences
Peak Points
Rational Function
Series
Squeeze Theorem
Subsequence
Tail

Squeeze Theorem for Sequences
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Definition
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Assume that for all , and
Then converges and .

Proof
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Choose a tolerance . Since , we can find an such that if , then and . But then if , this would mean that
implying that . This shows that converges and

Example of Use Crowdmark.pdf > page=6

Squeeze Theorem for Functions
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We can prove this using Sequential Characterization and Squeeze Theorem for Sequences. Similar to this Limits of a Function > Arithmetic Rules for Limits of Functions

Squeeze Theorem for Functions

Assume that three functions, , and , are defined on an open interval containing , except possibly at . Assume also that for each , except possibly , that

and that

Then exists and

Squeeze Theorem for Limits at
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Assume that . If
Then exists and equals .

Similarly,
Assume that . If
Then exists and equals

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Squeeze Theorem for Sequences

Squeeze Theorem for Functions

Squeeze Theorem for Limits at