If $\displaystyle f$ & $\displaystyle g$ are continuous on $\displaystyle R$, let

$\displaystyle S=\{x \in R: f(x) \geq g(x) \}$. Also let $\displaystyle (s_n) \subseteq S$. Suppose lim$\displaystyle (s_n)=s$, show that $\displaystyle s \in S$.

Well since f and g are continuous, we know

$\displaystyle \lim(s_n) = s$ then

$\displaystyle \lim(g(s_n)) = g(s)$ and $\displaystyle \lim(f(s_n)) = f(s)$

And $\displaystyle g(s)$ and $\displaystyle f(s)$ exists. I just don't know how to show $\displaystyle f(s) \geq g(s)$.

Would someone please help?