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
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Do you mean $[0,\infty)$? – velut luna Dec 03 '17 at 18:50
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No I mean [1,infinity). Task have 2 parts first was (0;1) and second [1; infinity). I mean you need to prove That function in uniform continuous on those compacts. – Семён Юрченко Dec 03 '17 at 18:52
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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*}
user284331
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There is a link in your question comment box, but if you take closer look of it, there is either derivative or integral taking place, so I think one must really have to go with anything beyond just elementary algebraic manipulations. – user284331 Dec 03 '17 at 19:46