TITLE: prove that the unit circle $x^2+y^2=1$ is a closed set
QUESTION [6 upvotes]: I need to prove that the unit circle $x^2+y^2=1$ is a closed set in $\mathbb{R}^2$ is closed using convergent sequence method.

REPLY [8 votes]: Consider the set $S=\{(x,y)\in\mathbb{R}^2\;|\; x^2+y^2=1\}$. Consider a sequence $(x_n,y_n)$ in S, s.t. it converges in $\mathbb{R}^2$.  We have $(x_n,y_n)\rightarrow (x,y)\in\mathbb{R}^2$ iff $x_n\rightarrow x\in\mathbb{R}$ and $y_n\rightarrow y\in\mathbb{R}$. This means that $x_n+y_n\rightarrow x+y$. Since $(x_n,y_n)\in S$ for all $n\in\mathbb{N}$ we have $x_n^2+y_n^2=1$ for all $n\in\mathbb{N}$. Therefore $x_n^2+y_n^2$ is a constant sequence (given equality to 1). Since $x_n^2+y_n^2\rightarrow x^2+y^2$ we get $x^2+y^2=1$. Therefore $(x,y)\in S$. This completes our proof.