# Show that ${(x,\sin(1/x)) : x∈(0,1]} ∪ {(0,y) : y ∈ [-1,1]}$ is closed in $\mathbb{R^2}$ using sequences

Convergent Sequence: A sequence $(x_n)$ in a metric space $X$ converges to $x_0$ if for every $\varepsilon>0$ there is $n_0\in\mathbb{N}$ such that $d(x_n, x_0)<\varepsilon$ for each $n≥n_0$.

Closed set: A set $F$ in a metric space $X$ is closed if and only if every convergent sequence in $F$ converges to a point in $F$.

Usual metric.

Ssvnormandysr1

211

## Answer

**Answers can be viewed only if**

- The questioner was satisfied and accepted the answer, or
- The answer was disputed, but the judge evaluated it as 100% correct.

Martin

480

The answer is accepted.

Join Matchmaticians Affiliate Marketing
Program to earn up to 50% commission on every question your affiliated users ask or answer.

- answered
- 323 views
- $10.00

### Related Questions

- Knot Theory, 3-colourbility of knots
- Taylor series/ Mclaurin Series
- Generalization of the Banach fixed point theorem
- [Precalculus] Sequences and series questions.
- Arithmetic Sequences Help
- Find sum of $n$ term, sum to infinity, least value for $n$
- using maclaurin series for tan(x) and equation for length of cable to prove that x=
- Undergrad algebraic topology proof