Proofs depend upon compactness, closed intervals are compact.
It can be done using uniform continuity, continuous functions on compact set are uniformly continuous. We can show then that continuous functions on compact sets are bounded. Then if bounded they have a LUB and GLB each of which are limit points of the range of the function. Thus, there is a highpoint and a lowpoint in the range.
Now that is no proof! But it tells you what is going on.