I am trying to show that $f(x)=e^{-x}$ is uniformly continuous on $[0,\infty)$ and not having much success. I'm attempting to use a modified version of the following result (which I found on Math Stack Exchange here: https://math.stackexchange.com/questions/172988/please-prove-uniform-continuity
), that if $f$ is continuous on $[a,\infty)$ and the limit of $f(x)=L$ as $x$ approaches infinity, then $f$ is uniformly continuous on $[a,\infty)$.
This is the work I've done so far. I really don't know if it's valid reasoning…
The first problem I'm encountering is that the book has not yet defined functional limits, so I can't really use the proof directly. We have, however defined sequential limits and the notion of a decreasing function. If I define a sequence $x_n=n-1$ for $n$$\in$$\mathbb{N}$, then as $n$ goes to infinity, $f(x_n)\rightarrow0$. For $\epsilon/2$, $\exists$N$\in$$\mathbb{N}$ s.th for all n $\geq$ N, |$f(x_n)|$ <$\epsilon/2$. Then, since $f(x)$ is a decreasing function, this inequality holds for all $x$$\geq$$x_N$.
Since $\mathcal{S}$=$[0,N+1]$ is closed and bounded, and hence a compact set in $\mathbb{R}$, and $f(x)=e^{-x}$ is continuous on $\mathcal{S}$, then by the Uniform Continuity Theorem, $f$ is uniformly continuous on $\mathcal{S}$.
So, if $x,y$ $\in$$\mathcal{S}$,$\forall$$\epsilon>0,$ $\exists$ $\delta>0$ s.th |$x-y$|<$\delta$ then |$f(x)-f(y)$|<$\epsilon$. Then (really not sure about this part), if $x, y$ $\in$$(N+1,\infty)$, for any $x$ and $y$ in this set, we have $|f(x)-f(y)|=|f(x)+(-f(y))|$$\leq|f(x)|+|f(y)|<\epsilon/2+\epsilon/2=\epsilon$. So $f$ is uniformly continuous on $(N+1,\infty)$. Provided this part is correct, I am having trouble with the case where $x$$\in[0,N+1]$ and $y$ $\in(N+1,\infty)$. Provided it's not correct, I'm having trouble on that part as well. Provided none of this is correct, well… I don't know.
Any help would be appreciated. Is this an appropriate way to approach the problem? Are there other ways?
Best Answer
For any $x>0$ and $y>0$ we have $$ \left\vert \mathrm{e}^{-x} - \mathrm{e}^{-y} \right\vert = \left\vert \mathrm{e}^{-\min(x,y)} \left(1 - \mathrm{e}^{-|y-x|}\right) \right\vert = \mathrm{e}^{-\min(x,y)} \left(1 - \mathrm{e}^{-|y-x|}\right) \leqslant \mathrm{e}^{-\min(x,y)} |y-x| $$ Hence for all $x>0, y>0$ such that $|x-y| < \delta$ $$ \left\vert \mathrm{e}^{-x} - \mathrm{e}^{-y} \right\vert < \delta $$ which verifies the direct definition of uniform continuity.