Prove that sinx/x uniformly continuous at x [1; infinity) by using epsilon delta

Question

My teacher told me to separate (1; infinity) into [1; delta(epsilon)];[delta(epsilon); infinity). Than prove first compact by Cantor’s theorem, and second by definition of uniform continuity. But how the process of choosing works? Do I chose literally any delta and epsilon or how? P.s. sorry for bad English

Answer 1

Given $\epsilon>0$, for any $x,y\in[1,\infty)$ such that $|x-y|<\epsilon/4$, assume that $y>x$, then \begin{align*} \left|\dfrac{\sin x}{x}-\dfrac{\sin y}{y}\right|&=\left|\dfrac{(y-x)\sin x+x\sin x-y\sin y-(x-y)\sin y}{xy}\right|\\ &\leq|x-y|\left|\dfrac{\sin x}{xy}\right|+|x-y|\left|\dfrac{\sin y}{xy}\right|+\left|\dfrac{x\sin x-y\sin y}{xy}\right|\\ &\leq 2|x-y|+|x-y|\cdot\dfrac{1}{y}\cdot|\sin\xi+\xi\cos\xi|,~~~~\xi<y\\ &\leq 2|x-y|+|x-y|(1+|\cos\xi|)\\ &\leq 4|x-y|. \end{align*}