if $f$ is bounded and continous, $f$ is uniformly continous

Question

S is closed, unbounded set on $\mathbb{R}$, and $f$ is bounded, continuous function on S. Suppose $\lim_{x\rightarrow\infty}f(x) \lim_{x\rightarrow-\infty}f(x)$ exists, Show that $f$ is uniformly continuous on S

Answer 1

Let $A=\lim_{x\to-\infty}f(x)$, $B=\lim_{x\to\infty}f(x)$, and let $\varepsilon>0$. Then there exist $x_0<y_0$ in $S$ such that $|f(x)-A|<\varepsilon/2$ if $x\leq x_0$ and $|f(y)-B|<\varepsilon/2$ if $y\geq y_0$ in $S$. Now, since $[x_0,y_0]$ is compact in $\mathbb{R}$, there exists $\delta>0$ such that $|f(x)-f(y)|<\varepsilon/2$ if $|x-y|<\delta$ in $S$. The conclusion follows easily.