1-prove-that-a-b-R-2-sinb-sina-b-a-2-let-give-the-sequence-x-0-0-and-x-n-1-a-1-2-sin-x-n-prove-that-for-m-n-x-m-x-n-a-2-n-1-3-prove-that-x-n-is-convergent-an

Question Number 33596 by abdo imad last updated on 19/Apr/18

1) p r o v e t h a t \forall (a, b) \in R^{2} ∣ s i n b - s i n a ∣⩽∣ b - a ∣

2) l e t g i v e t h e s e q u e n c e x_{0} = 0 a n d

x_{n + 1} = a + \frac{1}{2} s i n (x_{n}) p r o v e t h a t f o r m ⩾ n

∣ x_{m} - x_{n} ∣ ⩽ \frac{∣ a ∣}{2^{n - 1}}

3) p r o v e t h a t (x_{n}) i s c o n v e r g e n t a n d i t s l i m i t i s s o l u t i o n

o f t h e e q u a t i o n x = a + \frac{1}{2} s i n x .