I'm working on a university project studying numerical analysis and have hit a small snag. I have several theorems that deal with convergence of iterative procedures with respect to the absolute error at step $n$, i.e. $\epsilon_n = |x_n - x|$, where $x$ is the value being approximated by $x_n$.
My issue is that when I have written the programs to implement these methods I have used the error approximation of $\delta_n = |x_n - x_{n-1}|$. Testing has shown that these methods still converge, but I don't know how $\epsilon_n$ and $\delta_n$ have any connection to each other, and thus why $\delta_n$ is a valid error measure.
I've looked through several books without finding the answer and I want to be fairly rigorous with my perturbing choices in my project. In particular I an working with Newton's Method as one of those being used, and I can know that $\epsilon_n \le \epsilon_{n-1}$.