Calculus II · Unit 4A · lesson
The Squeeze Theorem for Sequences
Prove limits of bounded oscillating sequences with shrinking envelopes and the sequence form of the Squeeze Theorem.
Section overview
Sequences and their limitsWhat this section is building
Prove limits of bounded oscillating sequences with shrinking envelopes and the sequence form of the Squeeze Theorem.
A sequence converges when every sufficiently late term remains arbitrarily close to one finite target.
Use algebraic limits when a formula is explicit; use bounds and monotonicity when a recurrence hides the formula.
Reading a finite plot as proof or solving a recurrence's fixed-point equation before proving convergence.
Learning objectives
use upper and lower bounds to prove convergence when direct algebra does not reveal the limit.
The Squeeze Theorem for Sequences
Control can matter more than an exact formula
Some sequences contain oscillatory factors that never converge by themselves. If that oscillation is multiplied by a magnitude shrinking to zero, exact values become irrelevant. Bounding the sequence between two simpler sequences can force a limit even when the internal behavior remains complicated.
The sequence version of the Squeeze Theorem works exactly like its function counterpart. The important work is choosing bounds that hold for every sufficiently large integer. Absolute-value inequalities are especially efficient: proving with proves . This form also previews the error estimates used later for alternating and Taylor series.
Control can replace direct computation
An oscillating sequence may be difficult to simplify directly. The Squeeze Theorem ignores the exact internal motion and traps the sequence between two simpler sequences. If both walls close onto the same limit, the middle sequence has nowhere else to go.
The method is especially useful when a bounded factor is multiplied by something that shrinks to zero. The proof concerns the shrinking envelope, not the oscillation inside it.
Both boundaries eventually enter the same band
Once the lower and upper sequences lie inside an -band around , every trapped middle term lies inside as well. The formal proof simply chooses a cutoff large enough for both boundaries.
Sequence squeeze theorem
If for all sufficiently large , and and , then .
An oscillating numerator
Since ,
Both bounds approach zero, so
Bound the absolute value first
For , the numerator need not converge, but
Equivalently,
Both bounds approach zero, so .
The trap must actually close
The bounds do not imply convergence. The sequence satisfies them and still oscillates.
u4a-squeeze_theorem_for_sequences-01Evaluate .
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Prove .
Show using logarithms or known bounds.
Give an example where two bounds approach different limits and therefore do not determine the middle limit.
Explain the phrase "for all sufficiently large .
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