Proving the Nested Interval property of $\mathbb{R}$

Question

Let $\{I_n\}_1^\infty$ be a sequence of closed intervals $(I_n=[a_n,b_n])$ s.t.

(1) $I_n\supset I_{n+1} \:\text{for each}\: n\geq1$

(2) $\lim_{n\to\infty}(b_n-a_n)=0$

then $\exists!\:c$ s.t. $c\in I_n$ for each $n$, so $c=\bigcap_{n=1}^{\infty}I_n.$

Am I supposed to just show that $c\in\bigcap_{n=1}^{\infty}I_n$ or is does it require something more? If so, how would I go about it?

Answer 1

You are supposed to show there exists a unique number $c\in\mathbb R$ such that $$ \{ c \} = \bigcap_{n=1}^\infty I_n. $$ Note that the curly braces are missing in your question.

Hence, you have to show that there exists a $c\in \bigcap_{n=1}^\infty I_n$ and that there are no further elements in the intersection.