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I'm currently helping a friend on an assignment of his, but we are both stumbled on this exercise, I'm posting it here

We define a function ##f## which goes from ##\mathbb{R}## to ##\mathbb{R}## such that its argument maps as

$$

x \mapsto 1+\frac{1}{2}\sin\left(\frac{1}{x}\right)

$$

Show that

$$

\forall x,y \in\mathbb{R}\quad |f(x)-f(y)|\leq \frac{1}{2}\left|\frac{1}{x}-\frac{1}{y}\right|

$$

and deduce that

$$

\forall x,y \in [1,\infty) \quad |f(x)-f(y)|\leq \frac{1}{2}\left|x-y\right|

$$

That's it. The fact that it treats the limit of a general funtion with that argument is what is confusing, I tried to get a counterexample but haven't succeed. If anyone knows the first step on how to solve this or a hint it would be appreciated.

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# I Analysis of a general function with a specific argument

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