Let $f: \mathbb{R} \to \mathbb{R}$ be continuous. Then $G = \{ (x, f(x) ) : x \in \mathbb{R} \} $ is a closed set.
My try: Suppose $(z_n) = (x_n, f(x_n) ) $ is sequence in $G$ with limit $(x,y)$. We must show $(x,y) \in G$. Since $x_n \to x$ and since $f$ is continuous, then we must have that $f(x_n) \to f(x) $. Since limits are unique. Then $y = f(x) $. Is this enough to conlude that $(x,y) \in G $ ? Hence showing $G$ is closed ?
thanks