Prove that the sequence generated by {$x_k$} with arbitrary $x_0$ converges to the root of $f(x) = 0$

Question

Let $f(x)$ and $f'(x)$ be continuous in $(− ∞, ∞)$ and $M ≥ f'(x) ≥ m > 0$ for all $x ∈ (− ∞, ∞)$. Prove that the sequence ${x_k}$ generated by $x_{k +1} = x_k − λf(x_k)$, k = 0, 1, 2, ... with arbitrary $x_0$ converges to the root of $f(x) = 0$ for any choice of $λ$ in the interval $0 < λ < \frac{2}{M}$.

I thought of using methods from the order of convergence and Newton methods but I am clueless as to how I can approach this proof. Any suggestions and/ or examples will be deeply appreciated.

Answer 1

Just a Hint

Let $g(x)=x-\lambda f(x)$

$0<m\leq f'(x) \leq M \implies$

$0<m\leq \frac{1-g'(x)}{\lambda}\leq M$

$\implies \lambda m-1\leq -g'(x)\leq \lambda M-1$

the sequence converges if

$|g'(x)|<K<1 \implies \lambda M-1<1 \implies \lambda<\frac{2}{M}$.