Prove 1/n tends to zero using the epsilon definition.
Answer and explanation
Given ε>0, choose an integer N>1/ε. For n≥N, 0<1/n≤1/N<ε.
Credit the order of quantifiers, an explicit N, and a bound for every n≥N.
Subjects
Real analysis
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Sample exercises
Prove every convergent real sequence is bounded.
Answer and explanation
If a_n→L, eventually abs(a_n−L)<1, so the tail is bounded by abs(L)+1. Take the maximum of that bound and the finitely many initial absolute values.
Credit both tail control and the finite initial segment.
Which statement reverses the quantifiers incorrectly for convergence?
- There exists one N that works for every ε>0
- All sufficiently late terms lie within ε
- The cutoff may depend on ε
- For each ε>0 there is a suitable N
Answer and explanation
There exists one N that works for every ε>0
The needed cutoff generally grows as ε shrinks.
What is the smallest positive integer N such that 1/n<0.01 for every n≥N?
Answer and explanation
101
The strict inequality requires n>100, not n≥100.
Give a bounded real sequence that does not converge and justify it.
Answer and explanation
a_n=(−1)^n is bounded by 1 in absolute value, but its even and odd subsequences have limits 1 and −1.
Credit boundedness and a valid nonconvergence argument.
What extra property makes every Cauchy sequence converge in R?
- Completeness
- Countability
- A chosen decimal display precision
- All terms being integers
Answer and explanation
Completeness
Completeness prevents a Cauchy sequence from approaching a missing real-number limit.
Study notes
Sequence limits and epsilon proofs
For every ε>0 there exists N such that every n≥N satisfies the error bound.
Advanced universitySequence limits and epsilon proofs · revision notes
For every ε>0 there exists N such that every n≥N satisfies the error bound.
Advanced universityPointwise versus uniform convergence
The cutoff N may depend on both ε and the point x.
Advanced universityPointwise versus uniform convergence · revision notes
The cutoff N may depend on both ε and the point x.
Advanced universityCompactness and continuous functions
In R^n, compactness is equivalent to being closed and bounded.
Advanced universityCompactness and continuous functions · revision notes
In R^n, compactness is equivalent to being closed and bounded.
Advanced universityUniform error of a function sequence
A bound uniform in x proves more than pointwise convergence at a chosen x.
Advanced universityUniform error of a function sequence · formula reference
A bound uniform in x proves more than pointwise convergence at a chosen x.
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