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> Prove that $$\lim\_{x \rightarrow 0} \left[\frac{\arcsin x}{x}\right] = 1$$ and $$\lim\_{x \rightarrow 0} \left[\frac{\arctan x}{x}\right] = 0$$ where $[\,]$ represents the greatest integer function.
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For showing $\displaystyle\lim\_{x \rightarrow 0} \left[\frac{\sin x}{x}\right] = 0$ and $\displaystyle\lim\_{x \rightarrow 0} \left[\frac{\tan x}{x}\right] = 1$, I knew that $0 < \sin x < x < \tan x, x \in \left(0,\dfrac{\pi}{2}\right)$, hence that was easy but for $ \arcsin x$, I know no such inequality. Does $0 < \arctan x < x < \arcsin x$ hold for $x \in (0,1)$?