Problem:
"Show that if $f:\left[a,+\infty\right] \rightarrow \mathbb R$ is continuous and if $\lim_{x \rightarrow + \infty} f(x)$ exists, then $f$ is uniformly continuous".
I have a question about this problem.
My partial solution:
$\lim_{x\rightarrow +\infty}f(x) = L \Rightarrow \forall \varepsilon>0, \exists M>0$ such that $x \geq M \Rightarrow |f(x)-L|<\frac{\varepsilon}{2}$.
Given $x_1$,$x_2$ $\in$$[M, \infty)$$\Rightarrow |f(x_1)-L|<\frac{\varepsilon}{2}$ and $|f(x_2)-L|<\frac{\varepsilon}{2}$ $\Rightarrow |f(x_1)-L|<\frac{\varepsilon}{2}$ and $|L-f(x_2)|<\frac{\varepsilon}{2}$ $\Rightarrow |f(x_1)-L+L-f(x_2)|\leq |f(x_1)-L|+|L-f(x_2)|< \varepsilon$ $\Rightarrow |f(x_1) - f(x_2)|<\varepsilon$ $\Rightarrow$ $f$ is uniformly continuous on $[M, +\infty)$.
How can I extend this result towards $f$ uniformly continuous on $[a, +\infty]$?