Let \(x_{1}, x_{2}, x_{3}, \ldots\) be a sequence of real numbers (going on forever). For any integer \(n \geq 1\), define \(T_{n}\) to be the set \(\left\{x_{n}, x_{n+1}, \ldots\right\}\). (So, for example, \(T_{1}=\left\{x_{1}, x_{2}, x_{3}, \ldots\right\}\) and \(\left.T_{2}=\left\{x_{2}, x_{3}, x_{4}, \ldots\right\}.\right)\)

Assume that \(T_{1}\) has a lower bound. Deduce that for any \(n\), the set \(T_{n}\) has a GLB, and call it \(b_{n}\). Prove that \(b_{1} \leq b_{2} \leq b_{3} \leq \cdots\).

For the following sequences \(x_{1}, x_{2}, \ldots\), work out \(b_{n}\), and also work out the LUB of the set \(\left\{b_{1}, b_{2}, \ldots\right\}\) when it exists:

(a) \(x_{1}=1, x_{2}=2, x_{3}=3\), and in general \(x_{n}=n\),

(b) \(x_{1}=1, x_{2}=\frac{1}{2}, x_{3}=\frac{1}{3}\), and in general \(x_{n}=\frac{1}{n}\),

(c) \(x_{1}=1, x_{2}=2, x_{3}=1, x_{4}=2, x_{5}=1\), and so on, alternating between 1 and 2 .