I'm recently doing Numerical Analysis, more specifically about interpolation. In my book (A pretty old book which was not written in English), the author gives the following algorithm to find the solution of $f(x) = y$, where $y$ is given, $f(x)$ is an unknown function but is has a given data set $\{(x_i, y_i)\}_{i}$ with constant jump step $h = x_{i+1} - x_{i}$ and monotonically increase ($y_{i+1} > y_{i}$):
From the given data set $\{(x_i, y_i)\}_{i}$, using Newton forward interpolation formula, we have
$$y \approx y_0 + \Delta y_0\binom{t}{1} + \Delta^2 y_0\binom{t}{2} + \Delta^3 y_0\binom{t}{3} + \dots + \Delta^n y_0\binom{t}{n} \tag{1}$$
where $\Delta^n y_i = \Delta^{n-1}y_{i+1} - \Delta^{n-1}y_{i}$ and $t = \dfrac{x - x_0}{h}$. From $(1)$ we have
$$ t = \dfrac{1}{\Delta y_0} \left( y - y_0 - \Delta^2 y_0\binom{t}{2} - \Delta^3 y_0\binom{t}{3} - \dots \Delta^n y_0\binom{t}{n} \right) \tag{2} $$
Let $\varphi(t)$ be the R.H.S of $(2)$. Now the problem turns in to solving for t of the equation
$$t = \varphi(t)$$
Using loop with initial value $t_0 = \dfrac{y - y_0}{h}$ and the loop is defined by
$$t_k = \varphi(t_{k-1}),\, k\in\mathbb{N}^{\ast}$$
Keep using this loop until $t_k$ converge to some value $\overline{t}$, with error $\varepsilon$. Hence the solution of the problem $f(x) = y$ is $\overline{x} = x_0 + h\overline{t}$
The algorithm was understandable to me, however the author doesn't give any context about why the looping will eventually converge to a single value, and that single value is the solution. Can someone help me at this?