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