Notes on the Squeeze Theorem, Epsilon-Definition of Limit, and Monotone Sequences

Squeeze Theorem (intuition and formulation)

Core idea: If a sequence bn is trapped between two sequences an and cn that both converge to the same limit L, then bn also converges to L.
Formal statement (for sequences):
- There exists an index N such that for all n ≥ N,
  an \le bn \le c_n.
- If
  \lim{n \to \infty} an = \lim{n \to \infty} cn = L,
  then
  \lim{n \to \infty} bn = L.
Practical note: The inequalities need only hold eventually (for all n sufficiently large); early terms can be ignored for the limit.
Visual intuition: an and cn form two bounding rails converging to L; b_n lies in between and thus is forced to converge to L.

For a sequence (x_n) with limit L, the definition is:
- For every \epsilon > 0 there exists an index N ∈ \mathbb{N} such that for all n ≥ N,
  |x_n - L| < \epsilon.
Intuition: After some point, all terms lie within an \epsilon-neighborhood of L.
Consequence: If a sequence converges to L, it gets arbitrarily close to L and stays there after some index.

Define the three sequences:
- a_n = -\left(\tfrac{1}{2}\right)^n,
- b_n = \left(-\tfrac{1}{2}\right)^n,
- c_n = \left(\tfrac{1}{2}\right)^n.
Observation: For all n, we have an \le bn \le c_n.
- Reasoning: an = -tn, cn = tn with tn = (1/2)^n > 0, and bn = (-1)^n tn is either -tn or tn, so it lies between -tn and t_n.
Limits of the outer sequences:
- \lim{n \to \infty} an = -\lim_{n \to \infty} \left(\tfrac{1}{2}\right)^n = -0 = 0,
- \lim{n \to \infty} cn = \lim_{n \to \infty} \left(\tfrac{1}{2}\right)^n = 0.
By the Squeeze Theorem, the middle sequence converges to the same limit:
- \lim{n \to \infty} bn = 0.
Note on sign handling: If an ≤ bn ≤ cn and the outer limits are L, then the middle limit is also L. Also, for any sequence xn with limit L, one has
- \lim{n \to \infty} (-xn) = -\lim{n \to \infty} xn.
- Hence, in this example, since $(1/2)^n \to 0$, we have $- (1/2)^n \to 0$ as well, consistent with the squeeze.

Definition of increasing (strict):
- A sequence (an) is increasing if an < a_{n+1} \quad \text{for all } n \in \mathbb{N}.
Note on terminology: Some authors use "increasing" to mean strictly increasing, while others use "non-decreasing" for the non-strict case.
The phrasing in the transcript uses:
- Increasing: an < a{n+1} \text{ for all } n.
- Non-decreasing (to avoid ambiguity): an \le a{n+1} \text{ for all } n.
Practical remark: To avoid confusion, you may explicitly state whether you mean strict or non-strict monotonicity.

Links to the epsilon definition: The Squeeze Theorem relies on outer sequences having a known limit; the epsilon definition is the underlying formalization of limit behavior that makes these arguments rigorous.
Real-world relevance: In numerical analysis and approximation theory, bounding a difficult sequence between two simpler convergent ones is a common technique to establish convergence without computing the exact limit.
Philosophical/practical note: Rigorous limit statements require justification (existence of N or bounding inequalities) rather than intuition alone; the squeeze and monotonicity tools help ensure rigorous conclusions from fairly simple bounds.

Squeeze Theorem for sequences:
- If there exists N with an \le bn \le cn for all $n \ge N$ and \lim{n\to\infty} an = \lim{n\to\infty} cn = L, then \lim{n\to\infty} b_n = L.
Epsilon-definition of limit:
- For every \epsilon > 0 there exists N \in \mathbb{N} such that for all n \ge N,
  |x_n - L| < \epsilon.
Example sequences:
- an = -\left(\tfrac{1}{2}\right)^n,\ bn = \left(-\tfrac{1}{2}\right)^n,\ c_n = \left(\tfrac{1}{2}\right)^n.
- Inequality: an \le bn \le c_n\quad \text{for all } n.
- Limits: \lim{n\to\infty} an = \lim{n\to\infty} cn = 0\;\Rightarrow\; \lim{n\to\infty} bn = 0.
Monotone sequences:
- Increasing (strict): an < a{n+1}\quad \forall n.
- Non-decreasing: an \le a{n+1}\quad \forall n.