Question Number 33596 by abdo imad last updated on 19/Apr/18
$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\forall\left({a},{b}\right)\in{R}^{\mathrm{2}} \:\:\:\:\mid{sinb}\:−{sina}\mid\leqslant\mid{b}−{a}\mid \\ $$$$\left.\mathrm{2}\right){let}\:{give}\:{the}\:{sequence}\:\:{x}_{\mathrm{0}} =\mathrm{0}\:{and} \\ $$$${x}_{{n}+\mathrm{1}} ={a}\:+\frac{\mathrm{1}}{\mathrm{2}}{sin}\left({x}_{{n}} \right)\:{prove}\:{that}\:{for}\:{m}\geqslant{n} \\ $$$$\mid{x}_{{m}} \:−{x}_{{n}} \mid\:\leqslant\:\:\frac{\mid{a}\mid}{\mathrm{2}^{{n}−\mathrm{1}} } \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:\left({x}_{{n}} \right)\:{is}\:{convergent}\:{and}\:{its}\:{limit}\:{is}\:{solution} \\ $$$${of}\:{the}\:{equation}\:\:{x}\:=\:{a}\:+\frac{\mathrm{1}}{\mathrm{2}}\:{sinx}\:. \\ $$