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Secant Method - Error bound between kth iteration and root Watch

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    Hey guys.

    I know from the proof of the secant method that given the two initial values, x_{0} and x_{1}, are sufficiently close to the solution, x_{\ast}, then

    |x_{k+1} - x_{\ast}| < \frac{1}{2}|x_{k} - x_{\ast}|

    for k>0 where x_{k+1} is the solution after k solutions. This essentially says that the error bound between the actual solution and the k'th iterate is less than half the distance between the iterate before that and the actualy solution.

    How would i show using this fact (and possible other results) that

    |x_{k+1} - x_{\ast}| < |x_{k+1} - x_{k}|

    , which means that the error bound between the actual solution and the k'th iterate is less than the distance between the k'th iterate and the the k-1'th iterate.

    I'm very stumped on this question so if anyone has an idea i'd be extremely grateful

    Edit:
    I've been told that the Contraction Mapping Method could help me show this. As a reminder,



    It turns out that the error bound for this method is given by

    However, just like the error bound for the secant method that i gave in my question, this is not computable as the value of x_{\ast} is unknown.
    Now for the Contraction Mapping Method, the way they have manipulated the error bound is as shown,



    I can see the similarities between this and what i am trying to show with the Secant Method, but i don't understand exactly what they have done here, especially concerning the part under 'Now' on the above image.
    Can someone explain this to me and how i could use in my original problem?
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