Assume $ f(x) \in C^2[a,b] $ and $f(a)f(b)<0,f'(x)>0,f''(x)>0,\forall x \in[a,b]. $Prove the sequence $$ x_{n+1}=x_{n}-\frac{f(x_n)}{f'(x_n) } \quad x_1 \in[a,b]\text{ and } x_1 \text{is to the right of the zeropoint.}$$exist.And the limit is the root of $f(x)=0$
I know it is Newton tangent method, but how can I use sequence method like monotone boundary theorem to solve it?