  We present a new elementary proof of the existence of the least and the
greatest solutions to initial value problems in the conditions of Peano's
existence theorem. Our proof is based on a modification of Perron's method
which allows us to obtain quite easily the greatest solution as the solution
with biggest possible integral. In doing so, we simplify the usual proofs,
technically overloaded with lower (upper) solutions and/or related differential
inequalities. Moreover, those differential (and integral) inequalities, which
are interesting in their own right, can be quickly proven by means of known
techniques once we know that the greatest and the least solutions exist.
Summing up, we revert the usual approach from differential inequalities to
extreme solutions, getting a somewhat smoother presentation.
