I'm trying to show that the Nested Interval Theorem does NOT hold if the "first" interval in the sequence is unbounded.

So, $\displaystyle \big\{I_n\big\}_{n=1} ^\infty $ where $\displaystyle I_{n} = [a_{n}, b_{n}], a,b \in R$

with$\displaystyle I_{1} = [a_{1}, \infty)$, for example, implies that $\displaystyle \bigcap_{n}^{\infty} A_{n} = \varnothing $

The only thing that makes sense to me that I can think of is that $\displaystyle a_{1}, a_{2}, a_{3} ... a_{n}$ is an increasing sequence with no upper bound so eventually there is a possibility that for some large N $\displaystyle [a_{N}, b_{N}]$ becomes an empty set. And the intersection of any set with the empty set is the empty set.

Any suggestion or tips? Thanks