Given a sequence $(X_n)$ defined as follows: $X_1>0$ and $\forall n, X_{n+1}=\frac{1}{2}(X_n+\frac{b}{X_n})$
what do I need to think about when I see the notion of $X_{n+1}$? Should I think $X_{n+1}$ as a sequence? or just a way to defined the next element of the sequence $X_n$? I'm asking this because I saw that the limit of $X_{n+1}$ is equal to the limit of $X_n$ and the limit is defined for sequences. Why are the limits equal? Can I talk about the sequence $X_n$ and $X_{n+1}$ interchangeably? If so, why?
Furthermore, in a lecture I saw involving the same sequences above, in order to find out if the sequence $X_n$ is decreasing we evaluted the expression $X_{n+1}-X_n$ which is equal to $\frac{-(X_n)^2+b}{2X_n}$ However, because we don't have a formula for $X_n$ he developed $\frac{-(X_{n+1})^2+b}{2X_n}$ instead and then he concluded that $\forall n, b-(X_{n+1})^2 \le 0 $ from that he concluded that $\forall n \le2$, $b-(X_n)^2 \le 0$ why is that correct?
I'm quite confused - could you please help me to understand this fundamental concept?
Thank you very much for your time and help.