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

Question from a problem set. Definitions given:

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.


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