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The Limit Theorems (Part I)

MATH 4630 · Chapter 2: Sequences and Series

Having rigorously defined the convergence of sequences, we can now explore how limits interact with algebraic operations. This section establishes that the limiting process behaves predictably and intuitively with respect to arithmetic operations.

Why Do We Need These Theorems?

We've defined convergence precisely, but computing limits using $\varepsilon$-$N$ arguments for every sequence would be tedious. Consider:

$$\lim_{n \to \infty} \left(\frac{2n+1}{5n+4}\right)$$

Question: Must we use the definition directly, or can we manipulate limits algebraically?

Could we say:

$$\lim_{n \to \infty} \left(\frac{2n+1}{5n+4}\right) = \frac{\lim_{n \to \infty}(2n+1)}{\lim_{n \to \infty}(5n+4)} \quad ?$$

Bounded Sequences

Before establishing our main theorems, we need a preliminary result.

Definition: Bounded Sequence

A sequence $(x_n)$ is bounded if there exists a number $M > 0$ such that

$$ \quad \text{for all } n \in \mathbb{N}.$$

In other words, if a sequence $(x_n)$ is bounded, then every term of the sequence lies in the interval $[-M, M]$.

Theorem: Every Convergent Sequence is Bounded

Proof (Fill in the steps)

Suppose $(x_n) \to \ell$. Choose $\varepsilon = 1$. There exists $N \in \mathbb{N}$ such that whenever $n \geq N$, we have $|x_n - \ell| < 1$.

Step 1: For $n \geq N$

Show that $|x_n| < |\ell| + 1$.

Hint: Use the triangle inequality: $|x_n| = |x_n - \ell + \ell| \leq |x_n - \ell| + |\ell|$.

Step 2: For $n < N$

How many terms are there before the $N$-th term? Answer: There are many terms.

Step 3: Define the bound

Define $M = \max\{|x_1|, |x_2|, \ldots, |x_{N-1}|, |\ell| + 1\}$ and show that $|x_n| \leq M$ for all $n \in \mathbb{N}$.

The Algebraic Limit Theorem

Theorem: Algebraic Limit Theorem

Let $\lim\limits_{n \to \infty} a_n = a$ and $\lim\limits_{n \to \infty} b_n = b$. Then:

$\lim\limits_{n \to \infty}(c a_n) = $ for all $c \in \mathbb{R}$;
$\lim\limits_{n \to \infty}(a_n + b_n) = $ ;
$\lim\limits_{n \to \infty}(a_n b_n) = $ ;
$\lim\limits_{n \to \infty}\left(\dfrac{a_n}{b_n}\right) = $ , provided $b \neq 0$.

Proof of Part (i): Scalar Multiplication

Let $c=0$. Then the sequence $(ca_n)$ reduces to $(0,0,0,0,\ldots)$ and the theorem is trivial.

Let $c \neq 0$. We want to show $(ca_n) \to ca$.

Scratch work:

Let $\varepsilon > 0$. We need to find such that wherever , it follows that .

Now: $|ca_n - ca| = $ .

We know $(a_n) \to a$, so make: .

Formal proof:

Since $(a_n) \to a$, for the value , there exists $N \in \mathbb{N}$ such that whenever , it follows that

$$$$

Therefore, for all $n \geq N$:

Thus, $\lim_{n \to \infty}(ca_n) = ca$. $\square$

Proof of Part (ii): Sum of Sequences

We want to show $(a_n + b_n) \to a + b$.

Let $\varepsilon > 0$. We want: $|(a_n + b_n) - (a + b)| < \varepsilon$.

Scratch work using triangle inequality:

The "$\varepsilon/2$ trick": Make each piece less than $\varepsilon/2$.

Formal proof:

Let $\varepsilon >0$ be arbitrary. Since $(a_n) \to a$, there exists $N_1 \in \mathbb{N}$ such that whenever $n \geq N_1$, it follows that

$$|a_n - a| < $$

Since $(b_n) \to b$, there exists $N_2 \in \mathbb{N}$ such that whenever $n \geq N_2$, it follows that

$$|b_n - b| < $$

Let $N = $ . Then for $n \geq N$:

Therefore, $\lim\limits_{n \to \infty}(a_n + b_n) = a + b$. $\square$

Examples and Exercises

Example: Computing Limits Using the Algebraic Limit Theorem

Suppose $\lim\limits_{n \to \infty} a_n = 3$ and $\lim\limits_{n \to \infty} b_n = -2$. Find $\lim\limits_{n \to \infty}(5a_n + 2b_n)$.

Solution:

$$\lim_{n \to \infty}(5a_n + 2b_n) = 5 \lim_{n \to \infty} a_n + 2 \lim_{n \to \infty} b_n = 5(3) + 2(-2) = 15 - 4 = 11$$

We can apply the Algebraic Limit Theorem because both $(a_n)$ and $(b_n)$ converge.

Exercise 1

Suppose $\lim\limits_{n \to \infty} x_n = 4$. Find $\lim\limits_{n \to \infty}(3x_n^2 - 7x_n + 2)$.

Hint: Use the Algebraic Limit Theorem repeatedly.

Exercise 2

Let $(a_n)$ and $(b_n)$ be two sequences such that $\lim\limits_{n \to \infty} a_n = 0$ and $(b_n)$ is bounded. What can you conclude about $\lim\limits_{n \to \infty}(a_n b_n)$? Prove your answer.

Hint: Use the definition of "bounded" and the definition of convergence directly.

Key Takeaway

Limits commute with algebraic operations (addition, multiplication, scalar multiplication, and division) provided the underlying sequence limits exist and we avoid division by zero.

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