Boundedness of Sine X over X
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Theorem
Let $x \in \R$.
Then:
- $\size {\dfrac {\sin x} x} \le 1$
Proof
From Derivative of Sine Function, we have:
- $D_x \paren {\sin x} = \cos x$
So by the Mean Value Theorem, there exists $\xi \in \R$ between $0$ and $x$ such that:
- $\dfrac {\sin x - \sin 0} {x - 0} = \cos \xi$
From Real Cosine Function is Bounded we have that:
- $\size {\cos \xi} \le 1$
$\blacksquare$
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 16.3 \ (5)$