I really do not understand nested intervals. For example, one of my homework problems is: Let $x_0 \in \mathbb{R}$ and $x_{n+1}=\frac{1+x_n}{2}$ for all $n \in \mathbb{N}$. Prove that $\lim_{n \to \infty} x_n =1$. I think I might be able to do this problem if I better understood the nested interval stuff.
Edit: I plugged in some fake numbers and let $x_0=0$ and then $x_1=\frac{1}{2}$ and the number got closer and closer to 1, but I'm not sure how to prove that.
$$\begin{align} x_{n+1}&=\frac{1+x_n}2\\ \overbrace{x_{n+1}-1}^{u_{n+1}}&=\frac{(\overbrace{x_n-1}^{u_n})}2\\ u_{n+1}&=\frac{u_n}2\\ u_{n}&=\frac{u_{n-1}}2=\frac{u_{n-2}}{2^2}=\cdots=\frac{u_1}{2^{n-1}}\\ x_n-1&=\frac{x_1-1}{2^{n-1}}\\ x_n&=1+\frac{x_1-1}{2^{n-1}}\\ \lim_{n\to0}x_n&=1\qquad \blacksquare \end{align}$$