From my understanding of uniformly continuous functions, they will, by definition, map Cauchy sequences to Cauchy sequences (thus preserving the Cauchy sequence in its transformation). If a function does not possess this preservation then this function can be concluded to be not uniformly continuous on its domain.
I'm trying to use this fact to disprove the uniform continuity of the function $f(x)=xsin(x)$ on $[0,\infty)$. Using the periodic nature of sine, I'm trying to find a Cauchy sequence which will not map to a Cauchy sequence under $f$. Thus far I've only found non-cauchy sequences mapping to Cauchy sequences (e.g. $\{\pi(-1)^n\}_{n\in\mathbb{N}}$), which (in my interpretation of the property above) would not disprove uniform continuity. Does anyone have any hints (I'm looking more for guidance than an answer) on creating such a sequence (if this methodology of disproving is legitimate obviously)
The angle you're taking can be a legitimate way to disprove uniform continuity. As you noted, finding what non-Cauchy sequences map to is not fruitful and you want to find a Cauchy sequence mapping to a non-Cauchy sequence. But what needs to happen to a Cauchy sequence to make it non-Cauchy? Especially given that we're dealing with the set of nonnegative real numbers, Cauchy sequences are in fact convergent. Maybe you can use this to get more insight into what you're trying.