Prove $|x\sin\Big(\frac{1}{x}\Big)-y\sin\Big(\frac{1}{y}\Big)|\leq\sqrt{2|x-y|}$

Question

Prove that:

$$\Big|x\sin\Big(\frac{1}{x}\Big)-y\sin\Big(\frac{1}{y}\Big)\Big|\leq\sqrt{2|x-y|}$$

My attempt

I have prove that $$F(x,y)=\frac{f(x)-f(y)}{|x-y|^a}$$ $$f(x)=x\sin\Big(\frac{1}{x}\Big)$$have a bound if and only if $a\leq\frac{1}{2}$.

But I can't prove that $F(x)\leq\sqrt2$, if $a=\frac{1}{2}$.

So I want to get some help. Thank you.

Relative Questtion

I found some questons which is related to my question.

1

2

3