For gx cos x prove that the iteration xn1 gxn defines a con

For g(x) = cos x, prove that the iteration x_n+1:= g(x_n) defines a convergent sequence for an arbitrary initial value x_0.

Solution

Given g(x)=cosx and given iterative formula is x_n+1=g(x_n)

This iterative formula converges if |g(x)|1

Since g(x)=cosx and we know that |g(x)|=|cosx|1, for all real numbrs x,

We can conclude that the iterative formula x_n+1=g(x_n) converges,

for any choice of initial value x₀