# Thread: Having trouble with understanding continuity proof

1. ## Having trouble with understanding continuity proof

I'm having trouble with understanding one detail in this proof. I'd greatly appreciate if anyone can help.
Let $f:[0,\infty\) \rightarrow R$ be a continuous function. Suppose $\lim_{x \to \infty} f(x)= L$ exists and finite. Then $f$ is uniformly continuous on $[0,\infty\)$.
Proof: Let $\epsilon >0$. Since $f(x) \rightarrow L$ as $x \rightarrow \infty$, there exists $N>0$ st when $x>N$ we have $\mid f(x)-L \mid <\frac{\epsilon}{2}$. We know $f$ is uniformly continuous on $[0,2N]$ since $[0,2N]$ is compact and $f$ is continuous. Hence, there exists $\delta_1>0$ st for every $x,y \in [0,2N]$ , $\mid x-y \mid < \delta_1 \Rightarrow \mid f(x)-f(y)\mid < \epsilon$. Choose $\delta=min\{\delta_1,N/2\}$. Then any two points $x,y \in [0,\infty)$ satisfying $\mid x-y \mid <\delta$ are either both in $\[0,2N]$ or both in $[N,\infty)$. I understand the rest of the proof. The important detail that I don't get is this line
Choose $\delta=min\{\delta_1,N/2\}$. Then any two points $x,y \in [0,\infty)$ satisfying $\mid x-y \mid <\delta$ are either both in $\[0,2N]$ or both in $[N,\infty)$
I don't see why $x,y$ must be both in $[0,2N]$ or both in $[N,\infty)$. I guess it must have something to do with the distance between them less than delta, but I don't see it. I also don't see why we need the distance less than $N/2$.

2. ## Re: Having trouble with understanding continuity proof

if both points are not both either in [0,2N] or [N, $\infty$ ] this means one point has to be in [0,N) and the other point in (2N, $\infty$), meaning that their distance is greater than N. Now since $\delta$ is at least $\frac{N/2}$ (because of using min( $\{\delta, \frac{N}{2}\}$), this cannot be so both points have to be in [0,2N] or [N, $\infty$]