# Thread: Continuous Function Problem.

1. ## Continuous Function Problem.

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

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

Well since f and g are continuous, we know

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

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

Would someone please help?

2. Originally Posted by hockey777
If $f$ & $g$ are continuous on $R$, let

$S=\{x \in R: f(x) \geq g(x) \}$. Also let $s \subseteq S$. Suppose lim $(s_n)=s$, show that $s \in S$.
Look at that notation! There is sonething wrong with the symbols.
Please correct it.

3. Corrected.

I guess I should also add I know the $\lim(f(s_n)) \geq \lim(g(s_n))$

I'm struggling with what I should use to prove that fact.

4. Suppose that $g(s) > f(s)$ then let $\varepsilon = \frac{{g(s) - f(s)}}{2} > 0$.
$\left( {\exists N} \right)\left[ {n \geqslant N \Rightarrow \left| {g(s_n ) - g(s)} \right| < \varepsilon \,\& \,\left| {f(s_n ) - f(s)} \right| < \varepsilon } \right]$
Now you also know that $g(s_n ) \leqslant f(s_n )$.
From all of that a contradiction must follow.