Sequences, Limits, and Reading & Writing Proofs
The epsilon-N definition of a limit - your first genuinely rigorous definition - and habits for reading and writing proofs that others can check.
Leads to: This epsilon-N template becomes epsilon-delta continuity (Phase 2) and convergence in probability (Phase 7).
Learning Objectives
Click a status chip to cycle: Not started → In progress → Studied → Practiced → Needs review → Mastered.
- State the epsilon-N definition of the limit of a sequence and interpret each quantifier.
- Prove a specific limit directly by producing N as a function of epsilon.
- Explain why a bounded monotone sequence must converge.
- Read a proof critically and write one that a skeptical reader can verify line by line.
Key Vocabulary
- Sequence
- A function from the naturals to the reals, written (a_n); the ordered list a_1,a_2,….
- Convergence
- (a_n) converges to L if its terms are eventually arbitrarily close to L; written a_n → L.
- Epsilon-N definition
- ∀ε>0 ∃N such that n≥N ⇒ |a_n − L| < ε.
- Bounded sequence
- There is M with |a_n| ≤ M for all n.
- Monotone sequence
- Non-increasing or non-decreasing for all n.
- Divergence
- Failure to converge to any finite L; e.g. oscillation or growth without bound.
What a limit really says
Intuitively \(a_n\to L\) means ‘the terms get and stay close to \(L\)’. Rigor makes ‘close’ and ‘stay’ precise with the definition you spent 1.2 preparing for:
Read it as a game: an adversary picks a tolerance \(\varepsilon\); you must respond with a threshold \(N\) past which every term is within \(\varepsilon\) of \(L\). You win for every \(\varepsilon\) exactly when the limit holds. Smaller \(\varepsilon\) typically forces larger \(N\).
Monotone convergence: a limit without computing it
You can know a limit exists without a formula for it. The Monotone Convergence Theorem: a bounded, monotone sequence converges (to its supremum if increasing, its infimum if decreasing). This is why \((1+1/n)^n\) converges - it is increasing and bounded above - and its limit is named \(e\).
Reading and writing proofs
Reading a proof is active work: for each line ask ‘what justifies this - a definition, a hypothesis, or a previous line?’. Writing a proof is the mirror: make every justification explicit, define notation before using it, and state clearly where each hypothesis is used.
| Reader's question | What a good proof supplies |
|---|---|
| Why is this step allowed? | An explicit definition, theorem, or prior line |
| Where is hypothesis H used? | A visible appeal to H at the right moment |
| What is this symbol? | Introduced and typed before first use |
| Is the ‘arbitrary’ element really arbitrary? | No hidden special assumptions about it |
Interactive: watch terms fall inside the epsilon band
- Letting N depend on n. N may depend on ε only; it is chosen before n ranges past it.
- Proving |a_n − L| < ε for one convenient ε instead of every ε > 0.
- Assuming a bounded sequence converges - it need not (e.g. (−1)^n); monotonicity is also required.
- Writing ‘clearly’ or ‘obviously’ to paper over the one step you did not actually check.
- To prove a limit, work backwards from |a_n − L| < ε to an inequality on n; that reveals the N to pick.
- State ‘let ε > 0 be arbitrary’ first and ‘since ε was arbitrary’ last - the frame of every epsilon proof.
- When no formula for the limit is available, look for monotone + bounded and invoke Monotone Convergence.
- Read published proofs with a pen: reconstruct each skipped step yourself. That is retrieval practice for proof-writing.
Knowledge Check
Practical Exercise
Using the epsilon-N definition, prove that \(a_n=\dfrac{2n+1}{n}\) converges, and identify the limit.
Guess the limit: \(a_n=2+\tfrac1n\to 2\). Prove it. Fix \(\varepsilon\gt 0\). Then
We need \(1/n\lt \varepsilon\), i.e. \(n\gt 1/\varepsilon\). Choose any integer \(N\gt 1/\varepsilon\). Then for \(n\ge N\), \(|a_n-2|=1/n\le 1/N\lt \varepsilon\). Since \(\varepsilon\) was arbitrary, \(a_n\to 2\). □
Lesson Summary
Formula Sheet Additions
Retrieval Practice
Close the lesson and answer from memory before checking. This is deliberate, effortful recall - the single highest-yield study action.
A: N may depend only on ε (chosen before n is quantified). If N could depend on n the definition would be trivially satisfiable and meaningless.
A: A bounded monotone real sequence converges (increasing → its supremum, decreasing → its infimum); it proves existence of a limit even when no closed-form value is available, e.g. defining e.
Completion Checklist
- I can explain the core ideas in my own words
- I worked the derivations/examples by hand
- I completed the interactive workbench(es)
- I passed the knowledge check
Source References
This lesson synthesizes and paraphrases concepts from the sources below. No copyrighted text, problem sets, or solutions are reproduced. Return to the originals for full depth.
- Calculus, Vol. I (Tom Apostol, 2nd ed., 1967) foundational - Vol. I, Ch. 10 (sequences) - Rigorous treatment of sequences, the limit concept, and the completeness of the reals.
- Probability: Theory and Examples (Rick Durrett, 5th ed.) current - Ch. 1-2 background - Limits and convergence as the deterministic backbone for later modes of stochastic convergence.