If the interval are all disjoint then by density of the real numbers there must be real numbers in between the boundaries of the intervals. These numbers are not covered by your intervals and thus your intervals do not cover R.

The only subtlety is some sort of infinite sequence of intervals in the gaps but since these intervals are closed there are either going to be further gaps or they are going to share a boundary point which is disallowed. Not true if the intervals are open, but they're not.

You can do the work or turning this into a proof.