Let f and g be bounded functions from a nonempty set X into R.
Prove that if f(x) <= g(x) for all x in X, then inf f(X) <= inf g(X) and
sup f(X) <= sup g(X).
Assume supf(X)> supg(X)........................................... ................................................1
From the definition of the supremum we have:
for all x , if xεΧ then .................................................. ...........................................2
for all x, if xεΧ then .................................................. ............................................3
AND
for all ε>0 and hence for ε = supf(X)-supg(X)>0 there exists a yεf(X) such that :
supf(X)-(supf(X)-supg(X)< y or
supg(X) <y .................................................. ...........................................4
But since y belongs to f(X) y= f(x) and xεΧ and since for all xεΧ , (4) becomes:
supg(X)< g(x).............................................. ........................................5
And using (3) we end up with the contradiction:
supg(X) < supg(X)
the proof for the infemums is a similar one