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,
      a<em>n≤b</em>n≤cn.a<em>n \le b</em>n \le c_n.
    • If
      lim⁡<em>n→∞a</em>n=lim⁡<em>n→∞c</em>n=L,\lim<em>{n \to \infty} a</em>n = \lim<em>{n \to \infty} c</em>n = L,
      then
      lim⁡<em>n→∞b</em>n=L.\lim<em>{n \to \infty} b</em>n = 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.

Epsilon-definition of the limit (definition of convergence)

  • 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.

Example: Squeeze with three sequences

  • Define the three sequences:
    • an=−(12)n,a_n = -\left(\tfrac{1}{2}\right)^n,
    • bn=(−12)n,b_n = \left(-\tfrac{1}{2}\right)^n,
    • cn=(12)n.c_n = \left(\tfrac{1}{2}\right)^n.
  • Observation: For all n, we have a<em>n≤b</em>n≤cn.a<em>n \le b</em>n \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⁡<em>n→∞a</em>n=−lim⁡n→∞(12)n=−0=0,\lim<em>{n \to \infty} a</em>n = -\lim_{n \to \infty} \left(\tfrac{1}{2}\right)^n = -0 = 0,
    • lim⁡<em>n→∞c</em>n=lim⁡n→∞(12)n=0.\lim<em>{n \to \infty} c</em>n = \lim_{n \to \infty} \left(\tfrac{1}{2}\right)^n = 0.
  • By the Squeeze Theorem, the middle sequence converges to the same limit:
    • lim⁡<em>n→∞b</em>n=0.\lim<em>{n \to \infty} b</em>n = 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⁡<em>n→∞(−x</em>n)=−lim⁡<em>n→∞x</em>n.\lim<em>{n \to \infty} (-x</em>n) = -\lim<em>{n \to \infty} x</em>n.
    • Hence, in this example, since $(1/2)^n \to 0$, we have $- (1/2)^n \to 0$ as well, consistent with the squeeze.

Monotone sequences: Increasing and non-decreasing

  • 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: a<em>n<a</em>n+1 for all n.a<em>n < a</em>{n+1} \text{ for all } n.
    • Non-decreasing (to avoid ambiguity): a<em>n≤a</em>n+1 for all n.a<em>n \le a</em>{n+1} \text{ for all } n.
  • Practical remark: To avoid confusion, you may explicitly state whether you mean strict or non-strict monotonicity.

Connections to foundational principles

  • 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.

Quick recap of key formulas

  • Squeeze Theorem for sequences:
    • If there exists N with a<em>n≤b</em>n≤c<em>na<em>n \le b</em>n \le c<em>n for all $n \ge N$ and lim⁡</em>n→∞a<em>n=lim⁡</em>n→∞c<em>n=L,\lim</em>{n\to\infty} a<em>n = \lim</em>{n\to\infty} c<em>n = L, then lim⁡</em>n→∞bn=L.\lim</em>{n\to\infty} b_n = L.
  • Epsilon-definition of limit:
    • For every \epsilon > 0 there exists N∈NN \in \mathbb{N} such that for all n≥Nn \ge N,
      |x_n - L| < \epsilon.
  • Example sequences:
    • a<em>n=−(12)n, b</em>n=(−12)n, cn=(12)n.a<em>n = -\left(\tfrac{1}{2}\right)^n,\ b</em>n = \left(-\tfrac{1}{2}\right)^n,\ c_n = \left(\tfrac{1}{2}\right)^n.
    • Inequality: a<em>n≤b</em>n≤cnfor all n.a<em>n \le b</em>n \le c_n\quad \text{for all } n.
    • Limits: lim⁡<em>n→∞a</em>n=lim⁡<em>n→∞c</em>n=0  ⇒  lim⁡<em>n→∞b</em>n=0.\lim<em>{n\to\infty} a</em>n = \lim<em>{n\to\infty} c</em>n = 0\;\Rightarrow\; \lim<em>{n\to\infty} b</em>n = 0.
  • Monotone sequences:
    • Increasing (strict): a<em>n<a</em>n+1∀n.a<em>n < a</em>{n+1}\quad \forall n.
    • Non-decreasing: a<em>n≤a</em>n+1∀n.a<em>n \le a</em>{n+1}\quad \forall n.