Limits of Sequences

Introduction

Mathematical analysis studies the infinitesimal properties of functions. It is the entire theory whose formal axiomatic introduction requires a lot of time. If you are reading these pages, you are probably not looking for that. You want to gain or improve your skills in solving problems in calculus.

The aim of this set of notes is to introduce you to calculus without rigorously building the theory of mathematical analysis. Some of the proofs will be omitted or their details will be skipped. However, we will rigorously define the concept of limits of sequences. This way we will provide convincing arguments that the theory of limits and integration can be developed in the same way as number theory, combinatorics, and geometry. You will also get an insight of how proofs work in mathematical analysis. However, the main goal is to provide you with techniques for differentiation and integration and after this introductory chapter you may notice a more serious lack of details and rigor. This chapter is an exception. It is more mathematical analysis than it is calculus, but even if you think that that’s not what you want, please take your time to read this. And try to enjoy.

Motivation

We will define the limit of a sequence and establish some important properties of limits.

For motivational purposes we are going to give the following fake story of the history of number \( \pi \):

Million years ago people decided that it is important to find out the ratio between the perimeter and the diameter of a circle. The first calculations were not precise because people could make relatively small circles. They got that the ratio is \( 3 \). Let us denote by \( \rho_1 \) this first primitive ratio that the society has obtained. So, we got our first equation: \( \rho_1=3 \).

After some time, they built a better technology - i.e. the rope. The rope made it possible to make bigger circles and get a more precise ratio of the perimeter to the radius. And they got \( \rho_2=3.1 \).

After several years the price of rope decreased. Mathematicians got a big grant to buy a rope and they made their next scientific breakthrough: \( \rho_3=3.14 \). Year after year they improved this ratio. They obtained the sequence \[ \rho_1=3,\;\;\; \rho_2=3.1, \;\;\; \rho_3=3.14,\;\;\; \rho_4=3.142,\;\;\; \rho_5=3.1416,\;\;\; \rho_6=3.14159,\;\;\; \rho_7=3.141593,\;\;\; \dots\]

They figured out at that point that no society will ever live to see the final product: the exact ratio written as a number with a finite decimal expansion. The sequence \( (\rho_n)_{n=1}^{\infty} \) is going to be truly infinite sequence of numbers that are getting closer and closer to the real ratio. They gave the name \( \pi \) to the real ratio and said ``The sequence \( \left(\rho_n\right)_{n=1}^{\infty} \) converges to \( \pi \).’’

We want to define the notion of ``convergence.’’

Let us see a few more examples of sequences and further develop our intuition on convergence. The most trivial case is a constant sequence \( (a_n)_{n=1}^{\infty} \) defined as \( a_n=0 \). This sequence converges to \( 0 \). A typical divergent sequence occurs when there are two subsequences that approach to different limits. An example is the sequence \( (b_n)_{n=1}^{\infty} \) defined as \[ b_n=\left\{\begin{array}{ll} 0,&\mbox{ if }n \mbox{ is odd,}\\ 1,&\mbox{ if }n\mbox{ is even.}\end{array}\right.\]

Another example of divergent sequence is \( (d_n)_{n=1}^{\infty} \) defined as: \[ d_n=\left\{\begin{array}{ll} \frac1n,&\mbox{ if }n \mbox{ is odd,}\\ 1+\frac1n,&\mbox{ if }n\mbox{ is even.}\end{array}\right.\] Notice that the sequences \( \left(\frac1n\right)_{n=1}^{\infty} \) and \( \left(1+\frac1n\right)_{n=1}^{\infty} \) are both convergent.

Definition of the limit

The sequence whose elements are \( a_1 \), \( a_2 \), \( \dots \) will be referred to as \( (a_n)_{n=1}^{\infty} \).

Definition
 
A sequence \( (a_n)_{n=1}^{\infty} \) of real numbers is called convergent if there exists a real number \( L \) that satisfies:

For each \( \varepsilon> 0 \) there exists a positive integer \( n_0\in\mathbb N \) such that \( n\geq n_0 \) implies \( \left|a_n-L\right|< \varepsilon \).

The real number \( L \) is called the limit of the sequence \( (a_n)_{n=1}^{\infty} \) and we write this as \[ \lim_{n\to\infty}a_n=L.\]

Example
 
Assume that \( a_n=\left(3+\frac1n\right)^2 \). Prove that this sequence is convergent and find its limit.

Main properties of limits

Theorem 1 (Convergent sequences are bounded)
 
Assume that \( (a_n)_{n=1}^{\infty} \) is convergent. Then there exists a real number \( M \) such that \( |a_n|< M \) for all \( n\in\mathbb N \).

Theorem 2 (Squeeze Theorem)
 
Assume that \( (a_n)_{n=1}^{\infty} \), \( (b_n)_{n=1}^{\infty} \), and \( (c_n)_{n=1}^{\infty} \) are three sequences such that there exists \( n_0\in\mathbb N \) that satisfies \[ n\geq n_0\quad\quad\quad \mbox{implies}\quad\quad\quad a_n\leq b_n\leq c_n.\] If \( \displaystyle\lim_{n\to\infty}a_n=\lim_{n\to\infty} c_n=L \), then the sequence \( (b_n)_{n=1}^{\infty} \) is convergent and \( \displaystyle\lim_{n\to\infty}b_n=L \).

Theorem 3 (Limit of the sum)
 
Assume that \( (a_n)_{n=1}^{\infty} \) and \( (b_n)_{n=1}^{\infty} \) are two convergent sequences whose limits are \( \alpha \) and \( \beta \), respectively. Then the sequence \( (a_n+b_n)_{n=1}^{\infty} \) is convergent and its limit is \( \alpha+\beta \).

Theorem 4 (Limit of the product)
 
Assume that \( (a_n)_{n=1}^{\infty} \) and \( (b_n)_{n=1}^{\infty} \) are two convergent sequences whose limits are \( \alpha \) and \( \beta \), respectively. Then the sequence \( (a_n\cdot b_n)_{n=1}^{\infty} \) is convergent and its limit is \( \alpha\cdot \beta \).

Theorem 5 (Limit of the quotient)
 
Assume that \( (a_n)_{n=1}^{\infty} \) and \( (b_n)_{n=1}^{\infty} \) are two convergent sequences whose limits are \( \alpha \) and \( \beta \), respectively. Assume that \( b_n\neq 0 \) for all \( n \) and that \( \beta\neq 0 \). Then the sequence \( \left(\frac{a_n}{b_n}\right)_{n=1}^{\infty} \) is convergent and its limit is \( \frac{\alpha}{\beta} \).

A sequence \( (b_n)_{n=1}^{\infty} \) is called a subsequence of \( (a_n)_{n=1}^{\infty} \) if there exists an increasing sequence of positive integers \( i_1 \), \( i_2 \), \( \dots \), such that \( b_n=a_{i_n} \).

Theorem 6
 
If \( (a_n)_{n=1}^{\infty} \) is a convergent sequence with limit \( L \), then every subsequence of \( (a_n) \) is convergent and its limit is \( L \).

Theorem 7
 
If \( (a_n)_{n=1}^{\infty} \) is a convergent sequence of positive real numbers whose limit is \( L \), then the sequence \( (b_n)_{n=1}^{\infty} \) defined as \( b_n=\sqrt{a_n} \) is convergent as well and its limit is \( \sqrt L \).

The previous theorem can be written as \[ \lim_{n\to\infty} \sqrt{a_n}=\sqrt{\lim_{n\to\infty}a_n}.\] provided that \( (a_n) \) has non-negative terms and that it is convergent. We will later see that for a class of functions (called continuous functions) we have \( \displaystyle \lim_{n\to\infty} f(a_n)=f\left(\lim_{n\to\infty}a_n\right) \).

The following theorems allow us to establish the existence of limits even in situations when we do not see what the limit of the sequence is. They will imply the existence but they won’t tell us what is the limit. These theorems played a central role in the development of mathematical analysis. The ideas behind their proofs are far reaching. They lead to the theory of Banach spaces.

Theorem 8
 
If the sequence \( (a_n)_{n=1}^{\infty} \) is monotone and bounded, then it is convergent.

Theorem 9
 
For each bounded sequence \( (a_n)_{n=1}^{\infty} \) we define the sequences \( (\overline{a}_n)_{n=1}^{\infty} \) and \( (\underline{a}_n)_{n=1}^{\infty} \) in the following way: \begin{eqnarray*} \overline{a}_n&=& \sup\left\{a_n,a_{n+1},\dots \right\},\\ \underline{a}_n&=&\inf\left\{a_n,a_{n+1},\dots\right\}. \end{eqnarray*}
  • (a) The sequences \( (\overline{a}_n)_{n=1}^{\infty} \) and \( (\underline{a}_n)_{n=1}^{\infty} \) are convergent and their limits are called \( \displaystyle \limsup_{n\to\infty} a_n \) and \( \displaystyle \liminf_{n\to\infty} a_n \).

  • (b) There are subsequences \( (b_n)_{n=1}^{\infty} \) and \( (c_n)_{n=1}^{\infty} \) of \( (a_n)_{n=1}^{\infty} \) such that \[ \lim_{n\to\infty} b_n=\limsup_{n\to\infty} a_n\quad\quad\quad \mbox{and} \quad\quad\quad \lim_{n\to\infty} c_n=\liminf_{n\to\infty} a_n.\]

  • (c) The sequence \( (a_n)_{n=1}^{\infty} \) is convergent if and only if \[ \limsup_{n\to\infty}a_n=\liminf_{n\to\infty} a_n.\]

Theorem 10 (Cauchy’s Theorem)
 
Assume that the sequence \( (a_n)_{n=1}^{\infty} \) has the following property: For every \( \varepsilon> 0 \) there exists \( n_0 \) such that the inequality \( |a_n-a_m|< \varepsilon \) holds for any two natural numbers \( m \) and \( n \) that satisfy \( m\geq n_0 \) and \( n\geq n_0 \).

Then the sequence \( (a_n)_{n=1}^{\infty} \) is convergent.

The number \( e \)

Theorem 11 (Number \( e \))
 
Let \( (a_n)_{n=1}^{\infty} \) be the sequence defined as: \[ a_n=\left(1+\frac 1n\right)^n.\] This sequence is convergent.

Remark. The limit of the previous sequence is denoted by \( e \). Its approximate value is \( e\approx 2.718281828\dots \).


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