Calculate the degree of convergence I was studying about numerical methods to find root of functions. In the False Position Method, I came to the following claim:
$$\text{Degree of convergence } = P = \frac{1+\sqrt5}{2}$$
I cannot find any proof about it. Please help.
The claim is wrong. This rate of convergence applies only to the secant method. The pure vanilla regula falsi applied to monotone convex functions will develop a one-sided behavior, where only one side of the guard interval is moved. This can be easily checked on a diagram with paper and pencil. Convergence in this case is linear, order 1.

That is why the active point modifications were invented, the most simple being the Illinois variant. They restore superlinear convergence, but I'd think that a proof would be rather technical.
Here's a proof of the rate of convergence for a similar method.  Perhaps you can use a similar proof for your purposes.