Sequence Limits: Convergence, Limit Theorems, and Extensions

All flashcards

What is the epsilon–N definition of convergence for a real sequence?

For every $\varepsilon>0$, there is an $N$ such that $|x_n-L|<\varepsilon$ for every $n\ge N$.

What does the squeeze theorem establish for sequences?

If a sequence is eventually bounded above and below by sequences with the same limit, it has that limit too.

What is the monotone convergence theorem for real sequences?

Every real sequence that is both monotone and bounded converges.

What is a Cauchy sequence?

A sequence whose terms eventually become arbitrarily close to one another.

What distinguishes uniform from pointwise convergence?

Uniform convergence uses one threshold that works for every index; pointwise convergence allows the threshold to depend on the index.

Quiz questions

1. Why can a convergent real sequence have only one limit?Answer: Distinct proposed limits have disjoint sufficiently small neighborhoods, which cannot both contain all sufficiently late terms.
2. Suppose $a_n\le c_n\le b_n$ eventually, and both $a_n$ and $b_n$ converge to $L$. What follows?Answer: $c_n$ converges to $L$.
3. What does $x_n\to+\infty$ mean for a real sequence?Answer: For every real bound, all sufficiently late terms exceed that bound.
4. For $a_n\to A$ and $b_n\to B\ne0$, when does the quotient limit rule apply?Answer: When the quotient is defined for sufficiently large $n$; then $a_n/b_n\to A/B$.
5. Why does the double sequence $x_{n,m}=n/(n+m)$ have no double limit?Answer: Different relative growth rates of the two indices produce different limiting values.