I just watched a video about a problem stated as: "Find where $f(x)= x^7-1000$ intersects the $x$ axis. Find solution correct to 8 decimal places."
The author uses Newton's Method, repeating the iteration process using 8 decimal place approximations for $x_n$ until the point where the last approximation is the exact same as the previous one, namely $2.68269580$.
He then concludes that the equality of the last 2 approximations means the approximation is correct to 8 decimal places.
How/Why does that work?
I don't understand how the fact that the last approximation is the same as the previous one guarantees that the digits won't change anymore up to that number of decimal places.
I'm assuming it has to do with the process, and that in this case we know (please correct me if wrong) that N's method guarantees that any two subsequent approximations are closer than any other two subsequent approximations which appear earlier in the approximation sequence.
But even if that's provably true, couldn't there be a case for example where we get in our approximation sequence: $2.--------777$, and then $2.--------999$, at which point we would conclude that our final approximation is correct up to 8 decimal places, even though an increase in just 0.00000000001 in the next approximation would be enough to change at least the digit in our 8th decimal place?
Also, from this question: Calculate cosh 1 correct to 6 decimal places., one answer states "You will only have to develop the series for a few steps until the first significant digits will not change any more."
But how do we know that's true? How do we know that once a certain digit stops changing it'll never change again?