Proving nested interval theorem from least upper bound property The following line appeared in the following proof: LINK
Let $X=⋂_{i\in I}[a_i,b_i]$. If $I=∅$, then $X=\mathbb R≠∅$.
What is the reason for this line? Is it just a definition?
The rest of the proof makes full sense.
Thanks
Jason
 A: He is saying that the nested interval property works even if the set of nested intervals is indexed by the empty set, $\emptyset$. The intersection over the empty set is always the space you are working in because the elements of an intersection are those elements that appear in every set being intersected. As there are no sets for the elements to not appear in, they appear in every set.
A: Let $x\in X$. 
Do we have $x\in\bigcap_{i\in I}\left[a_{i},b_{i}\right]$ here? 
If not then there
must be some $i\in I$ such that $x\notin\left[a_{i},b_{i}\right]$.
However, if $I=\emptyset$ the no such $i$ exists.
So we find that $x\in\bigcap_{i\in \emptyset}\left[a_{i},b_{i}\right]$  is true for each $x\in X$. 
Under the convention that we are only looking at subsets of $X$ here we end up with $$\bigcap_{i\in \emptyset}\left[a_{i},b_{i}\right]=X$$
If this convention is 'dropped' then we come into trouble. This because it can also be shown that $x\in\bigcap_{i\in \emptyset}\left[a_{i},b_{i}\right]$ is vacuously true for any $x\notin X$
