This is a beta course, so its structure, chapters, and examples may continue to change.

Properties of Limits

Limits have some important properties, which are the basis for limit computation and proof.

Uniqueness

Uniqueness of a limit

If the sequence $\{a_n\}$ converges, then its limit is unique.

Meaning: a convergent sequence cannot tend to two different numbers at the same time.

Boundedness

Boundedness of a convergent sequence

If the sequence $\{a_n\}$ converges, then $\{a_n\}$ is necessarily bounded.

Meaning: all terms of a convergent sequence lie within some range.

Note: boundedness is a necessary condition for convergence, but not a sufficient one.

Counterexample: $a_n = (-1)^n$ is bounded but does not converge.

Sign Preservation

The sign-preserving property of limits

If $\lim_{n \to \infty} a_n = A > 0$ , then there exists an $N$ such that when $n > N$ , $a_n > 0$ .

Meaning: if the limit is positive, then from some term onward, all terms are positive.

The Squeeze Theorem

The squeeze theorem

If there exists an $N$ such that when $n > N$ , $b_n \leq a_n \leq c_n$ , and $\lim_{n \to \infty} b_n = \lim_{n \to \infty} c_n = A$ , then $\lim_{n \to \infty} a_n = A$ .

Intuitive understanding: if $a_n$ is “sandwiched” between two sequences tending to the same limit, then $a_n$ also tends to this limit.

Worked example:

Find $\lim_{n \to \infty} \frac{\sin n}{n}$

Since $-1 \leq \sin n \leq 1$ , we have $-\frac{1}{n} \leq \frac{\sin n}{n} \leq \frac{1}{n}$

And $\lim_{n \to \infty} (-\frac{1}{n}) = \lim_{n \to \infty} \frac{1}{n} = 0$

By the squeeze theorem, $\lim_{n \to \infty} \frac{\sin n}{n} = 0$

The Monotone Convergence Theorem

The monotone convergence theorem

A monotone and bounded sequence necessarily converges.

More specifically:

A monotonically increasing sequence with an upper bound necessarily converges
A monotonically decreasing sequence with a lower bound necessarily converges

Application: this theorem is often used to prove that a sequence converges, even when we do not know the value of the limit.

Practice Problems

Exercise 1

Use the squeeze theorem to find: $\lim_{n \to \infty} \frac{\cos n}{n^2}$

Reference Answer(1 个标签)

properties of sequence limits

Solution:

Since $-1 \leq \cos n \leq 1$ , we have $-\frac{1}{n^2} \leq \frac{\cos n}{n^2} \leq \frac{1}{n^2}$

And $\lim_{n \to \infty} (-\frac{1}{n^2}) = \lim_{n \to \infty} \frac{1}{n^2} = 0$

By the squeeze theorem, $\lim_{n \to \infty} \frac{\cos n}{n^2} = 0$

Exercise 2

Prove that the sequence $a_n = \frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{2n}$ converges.

Reference Answer(1 个标签)

properties of sequence limits

Solution:

First prove monotonicity: $a_{n+1} - a_n = \frac{1}{2n+1} + \frac{1}{2n+2} - \frac{1}{n+1} = \frac{1}{2n+1} + \frac{1}{2n+2} - \frac{2}{2n+2} = \frac{1}{2n+1} - \frac{1}{2n+2} > 0$

So $\{a_n\}$ is monotonically increasing.

Then prove boundedness: $a_n = \frac{1}{n+1} + \cdots + \frac{1}{2n} < \frac{n}{n+1} < 1$

By the monotone convergence theorem, $\{a_n\}$ converges.

Exercise 3

Use the squeeze theorem to find: $\lim_{n \to \infty} \frac{1 + 2 + 3 + \cdots + n}{n^2}$

Reference Answer(1 个标签)

properties of sequence limits

Solution:

$\frac{1 + 2 + \cdots + n}{n^2} = \frac{\frac{n(n+1)}{2}}{n^2} = \frac{n+1}{2n} = \frac{1}{2} + \frac{1}{2n}$

$\lim_{n \to \infty} \left(\frac{1}{2} + \frac{1}{2n}\right) = \frac{1}{2}$

Or use the squeeze theorem: $\frac{n \cdot n}{n^2} \leq \frac{1 + 2 + \cdots + n}{n^2} \leq \frac{n \cdot n}{n^2}$

But this is not right. The correct squeezing: $\frac{n \cdot 1}{n^2} \leq \frac{1 + 2 + \cdots + n}{n^2} \leq \frac{n \cdot n}{n^2}$

i.e., $\frac{1}{n} \leq \frac{1 + 2 + \cdots + n}{n^2} \leq 1$

This range is too wide. Direct computation is simpler.

Answer: $\frac{1}{2}$

Summary

Symbols Used in This Article

符号	类型	读音/说明	在本文中的含义
$a_n, b_n, c_n$	数学符号	a/b/c-sub-n	Sequences
$A$	数学符号	A	The limiting value of a sequence
$N$	数学符号	N	The term threshold
$(-1)^n$	数学符号	minus one to the n	The alternating sign
$\sin n, \cos n$	数学符号	sine/cosine of n	Trigonometric functions
$\leq$	数学符号	less than or equal to	Less than or equal to
$\infty$	数学符号	infinity	Infinity

中英对照

中文术语	英文术语	音标	说明
唯一性	uniqueness	/juːˈniːknəs/	A limit has only one value
有界性	boundedness	/ˈbaʊndɪdnəs/	The sequence lies within some range
夹逼定理	squeeze theorem	/skwiːz ˈθɪərəm/	Also called the sandwich theorem
单调收敛定理	monotone convergence theorem	/ˈmɒnətəʊn kənˈvɜːdʒəns ˈθɪərəm/	Monotone and bounded implies convergence
保号性	sign-preserving property	/saɪn prɪˈzɜːvɪŋ ˈprɒpəti/	The sign of the limit agrees with the terms

The Definition of Limits

Operations on Limits