# Convergence of a sequence in a metric space

• Ted123
In summary, the problem is asking to show that a sequence (x_n) converges to a limit x in the metric space (\mathbb{R}, d_2) if and only if it converges in the metric space (\mathbb{R},d_1). This can be shown by using the continuity of arctan and the fact that d_2 is a metric with the property that d_2(x,y) < \pi for all x,y \in \mathbb{R}.

## Homework Statement

For $x,y \in\mathbb{R}$ define a metric on $\mathbb{R}$ by $$d_2(x,y) = |\tan^{-1}(x) - \tan^{-1}(y) |$$ where $\tan^{-1}$ is the principal branch of the inverse tangent, i.e. $\tan^{-1} : \mathbb{R} \to (-\pi/2 ,\pi/2)$.

If $(x_n)_{n\in\mathbb{N}}$ is a sequence in $\mathbb{R}$ and $x\in\mathbb{R}$, show that $x_n \to x$ as $n\to\infty$ in $(\mathbb{R} ,d_1)$ where $d_1$ is the standard metric $d_1(x,y)=|x-y|$ if and only if $x_n \to x$ as $n\to\infty$ in $(\mathbb{R} ,d_2)$.

## The Attempt at a Solution

$x_n\to x$ in $(\mathbb{R},d_2) \iff d_2(x_n,x)\to 0$

$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\, \iff |\tan^{-1}(x_n) - \tan^{-1}(x) | \to 0$

$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\, \iff \tan^{-1}(x_n) \to \tan^{-1}(x)$

$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\, \iff x_n \to x$ pointwise (since $\tan^{-1} : \mathbb{R} \to (-\pi/2 ,\pi/2)$ is continuous)

$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\, \iff |x_n - x| \to 0$

$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\, \iff d_1(x_n,x) \to 0$

$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\, \iff x_n\to x$ in $(\mathbb{R},d_1)$

I'm not sure whether the pointwise bit in the middle is correct (as that seems to imply pointwise convergence and convergence in a metric is the same, when it isn't) and that is the crucial step!

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Ted123 said:
$\tan^{-1}(x_n) \to \tan^{-1}(x)\iff x_n \to x$ pointwise (since $\tan^{-1} : \mathbb{R} \to (-\pi/2 ,\pi/2)$ is continuous)

I'm not sure if this addresses the pointwise vs. metric convergence you ask about, but there is a problem in your justification here. You're right that if $\lim_{n\to \infty} x_n = x$, then $\lim_{n \to \infty} \tan^{-1}(x_n) = \tan^{-1}(x)$ by the continuity of arctan, but the converse isn't always true for a given continuous function. For instance, take $f(x) = x^2$ and $x_n = -1$ for all n. Then $\lim_{n \to \infty} f(x_n) = 1 = f(1)$, but $\lim_{n \to \infty} x_n \neq 1$.

But this problem doesn't arise for arctan. Can you think what property of arctan makes this implication true?

spamiam said:
I'm not sure if this addresses the pointwise vs. metric convergence you ask about, but there is a problem in your justification here. You're right that if $\lim_{n\to \infty} x_n = x$, then $\lim_{n \to \infty} \tan^{-1}(x_n) = \tan^{-1}(x)$ by the continuity of arctan, but the converse isn't always true for a given continuous function. For instance, take $f(x) = x^2$ and $x_n = -1$ for all n. Then $\lim_{n \to \infty} f(x_n) = 1 = f(1)$, but $\lim_{n \to \infty} x_n \neq 1$.

But this problem doesn't arise for arctan. Can you think what property of arctan makes this implication true?

I've just realized that the question points out that $d_2$ is a metric with the property that $d_2(x,y)< \pi$ for all $x,y\in\mathbb{R}$ - does this help? I can't see what property of arctan makes this implication true.

## What is the definition of convergence in a metric space?

Convergence of a sequence in a metric space refers to the behavior of a sequence as its terms approach a specific limit point. In other words, as the terms in the sequence get closer and closer to a particular value, the sequence is said to be converging.

## How is convergence in a metric space different from convergence in a real number system?

In a metric space, convergence is determined by the distance between the terms in the sequence and the limit point. In contrast, in a real number system, convergence is determined by the absolute value of the difference between the terms and the limit point.

## What is the importance of convergence in a metric space?

Convergence in a metric space is a fundamental concept in analysis and is used to study the behavior of sequences and series. It allows us to determine whether a sequence has a limit and to make predictions about its future behavior.

## What are the conditions for a sequence to converge in a metric space?

A sequence in a metric space converges if and only if it satisfies two conditions: 1) the terms in the sequence get arbitrarily close to the limit point, and 2) the distance between the terms and the limit point becomes smaller as the sequence progresses.

## Can a sequence in a metric space converge to more than one limit point?

No, a sequence in a metric space can only converge to one limit point. If a sequence has multiple limit points, it is said to be divergent.