While solving an exercise leaflet, I got stuck on the following :
The equation $x^3 - 2x - 5 = 0$ has exactly one real solution $x_s$ and it belongs to the interval $(2,3)$. Show thoroughly that, if $x_0 >=1$ then the sequence $(x_k)$ that is given by the method Newton-Raphson, converges at the solution $x_s$.
My mind tells me that we shall show that the sequence $(x_k), k>=1$ is decreasing and that it's down bounded (Sorry about the term "down", but I do not know how to properly say it in English, so I translated it boldly from Greek (κάτω φραγμένη).
First of all I started by finding the sequence, which is :
$x_k = x_{k-1} - \frac{x_{k-1}^3 - 2x_{k-1} - 5}{3x_{k-1}^2 -2}$
By finding that it's decreasing and bounded, we can apply the Monotone Convergence Theorem and prove what we need, but I my mind is stuck and I cannot prove that the sequence is decreasing or bounded at this point. Also, I have a question in my mind about how we'll end up showing that the sequence converges to the solution exactly.
Any help will be appreciated ! I'd love and it would help me tons if someone could provide a thorough explanation. Thanks in advance !