> [!NOTE] > For all $n \geq 1, x_{1},x_{2},\dots,x_{n}\in \mathbb{R},$ $| \sin (x_{1}+x_{2}+\dots+x_{n})| \leq |\sin x_{1}| + | \sin x_{2}| +\dots+ | \sin x_{n}| \tag{1}$ ###### Proof $(1)$ is clearly holds true for $n=1.$ Suppose $(1)$ holds true $n=k.$ Then $\begin{align} |\sin (x_{1}+x_{2}+\dots+x_{k+1})| &= |\sin(x_{1}+x_{2}+\dots+\sin x_{k}) \cos(x_{k+1}) + \sin (x_{k+1}) \cos(x_{1}+x_{2}+\dots+x_{k}) | \\ & \leq |\sin (x_{1}+x_{2}+\dots+x_{k})| |\cos x_{k+1}| + |\sin x_{k+1}| |\cos(x_{1}+x_{2}+\dots+x_{k})| \\ &\leq |\sin (x_{1} + x_{2} +\dots+x_{k})| + |\sin x_{k+1}| \\ & \leq |\sin x_{1}| + | \sin x_{2}| +\dots+ | \sin x_{k}| + |\sin x_{k+1}| \end{align}$thus by [[Induction Principle|mathematical induction]], $(1)$ holds true for all $n$ as above.