Sequences of Vectors

Section 3.1 Sequences of Vectors

Given vectors

x_{n} = (x_{1 n}, \dots, x_{N n})

and

x = (x_{1}, \dots, x_{N})

R^{N}

we say that $x_{n}$ converges to $x$ as $n$ goes to infinity if all components

x_{i n}

x_{n}

converge to the corresponding component

x_{i}

x .

In symbols

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\begin{matrix} (3.1) & x_{n} \overset{n \to \infty}{\to} x ⟺ x_{i n} \overset{n \to \infty}{\to} x_{i} for all i = 1, \dots, N . \end{matrix}

We say that

x

is the limit of the sequence

(x_{n}),

and write

🔗

lim_{n \to \infty} x_{n} = x .

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As in case of sequences of numbers if the limit exists. If a sequence of real numbers

(t_{n})

converges to

t

then we know that

| t_{n} - t | \overset{n \to \infty}{\to} 0

and vice versa. A similar fact is true for sequences of vectors as the following proposition shows.

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Proposition 3.1. Convergence of sequences of vectors.

Let

(x_{n})_{n \in N}

be a sequence of vectors in

R^{N} .

Then

🔗

lim_{n \to \infty} x_{n} = x if and only if lim_{n \to \infty} ‖ x_{n} - x ‖ = 0 .

🔗

Proof.

By definition of the norm we have

\begin{matrix} (3.2) & ‖ x_{n} - x ‖ = (\sum_{i = 1}^{N} | x_{i n} - x_{i} |^{2})^{1 / 2} . \end{matrix}

x_{n} \overset{n \to \infty}{\to} x

then by (3.1) we know that

x_{i n} \overset{n \to \infty}{\to} x_{i}

for every

i = 1, \dots, N,

and thus every term on the right hand side of (3.2) goes to zero. This shows that

‖ x_{n} - x ‖ \overset{n \to \infty}{\to} 0

x_{n} \overset{n \to \infty}{\to} x .

Now assume that

‖ x_{n} - x ‖ \overset{n \to \infty}{\to} 0 .

Observe that every term in the sum on the right hand side of (3.2) is non-negative. Hence

0 \leq | x_{i n} - x_{i} | \leq (\sum_{i = 1}^{N} | x_{i n} - x_{i} |^{2})^{1 / 2} = ‖ x_{n} - x ‖

for every

i = 1, \dots, N .

By assumption

‖ x_{n} - x ‖ \overset{n \to \infty}{\to} 0,

and thus by the “squeezing lemma”

| x_{i n} - x_{i} | \overset{n \to \infty}{\to} 0

for all

i = 1, \dots, N .

This shows that thus

x_{i n} \overset{n \to \infty}{\to} x_{i}

for all

i = 1, \dots, N

as required.

As for sequences in

R

we can give an

ε

-characterisation of convergence. It can also be used as a definition of convergence of a sequence. The proof is similar to the proof of the above proposition and left as an exercise.

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Proposition 3.2.

A sequence

(x_{n})

R^{N}

converges to

x_{0} \in R^{N}

if and only if for every

ε > 0

there exists

n_{0} \in N

such that

‖ x_{n} - x_{0} ‖ < ε

for all

n > n_{0} .

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Limits of vectors have the same properties as limits of sequences of real numbers.

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Proposition 3.3.

Suppose that

(x_{n}),

(y_{n})

are sequences in

R^{N}

with limits

x

and

y,

respectively. Moreover, assume that

(α_{n})

is a sequence in

R

with limit

α .

Then

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$lim_{n \to \infty} (x_{n} + y_{n}) = lim_{n \to \infty} x_{n} + lim_{n \to \infty} y_{n} = x + y$
$lim_{n \to \infty} α_{n} x_{n} = (lim_{n \to \infty} α_{n}) (lim_{n \to \infty} x_{n}) = α x$
$lim_{n \to \infty} (x_{n} \cdot y_{n}) = (lim_{n \to \infty} x_{n}) \cdot (lim_{n \to \infty} y_{n}) = x \cdot y$
If $N = 3$ then $lim_{n \to \infty} (x_{n} \times y_{n}) = (lim_{n \to \infty} x_{n}) \times (lim_{n \to \infty} y_{n}) = x \times y$