Library / General Topology / Chapter IV / § 5
Proposition 12
Tag 0BRR, permanent. Cite it as [Bourbaki, Tag 0BRR].
General Topology, chapter IV, § 5. Real-valued functions
Let f f f g g g X X X A A A X X X
(i) We have
sup x ∈ A ( f ( x ) + g ( x ) ) ≤ sup x ∈ A f ( x ) + sup x ∈ A g ( x ) , sup x ∈ A f ( x ) + inf x ∈ A g ( x ) ≤ sup x ∈ A ( f ( x ) + g ( x ) ) ,
\begin{align*}
&\sup_{x \in A} (f(x) + g(x)) \leq \sup_{x \in A} f(x) + \sup_{x \in A} g(x), \\
&\sup_{x \in A} f(x) + \inf_{x \in A} g(x) \leq \sup_{x \in A} (f(x) + g(x)),
\end{align*}
x ∈ A sup ( f ( x ) + g ( x )) ≤ x ∈ A sup f ( x ) + x ∈ A sup g ( x ) , x ∈ A sup f ( x ) + x ∈ A inf g ( x ) ≤ x ∈ A sup ( f ( x ) + g ( x )) , whenever both sides of these inequalities are defined.
(ii) If f ( x ) f(x) f ( x ) g ( x ) g(x) g ( x ) ≥ 0 \geq 0 ≥ 0 x ∈ A x \in A x ∈ A
(19)
sup x ∈ A ( f ( x ) g ( x ) ) ≤ sup x ∈ A f ( x ) sup x ∈ A g ( x ) ,
\sup_{x \in A} (f(x)g(x)) \leq \sup_{x \in A} f(x) \sup_{x \in A} g(x),
x ∈ A sup ( f ( x ) g ( x )) ≤ x ∈ A sup f ( x ) x ∈ A sup g ( x ) , (20)
sup x ∈ A f ( x ) inf x ∈ A g ( x ) ≤ sup x ∈ A ( f ( x ) g ( x ) )
\sup_{x \in A} f(x) \inf_{x \in A} g(x) \leq \sup_{x \in A} (f(x)g(x))
x ∈ A sup f ( x ) x ∈ A inf g ( x ) ≤ x ∈ A sup ( f ( x ) g ( x )) whenever both sides of these inequalities are defined.
(iii) If f ( x ) ≥ 0 f(x) \geq 0 f ( x ) ≥ 0 x ∈ A x \in A x ∈ A (21)
sup x ∈ A ( 1 / f ( x ) ) = 1 / inf x ∈ A f ( x )
\sup_{x \in A} (1/f(x)) = 1/\inf_{x \in A} f(x)
x ∈ A sup ( 1/ f ( x )) = 1/ x ∈ A inf f ( x ) (putting 1 / 0 = + ∞ 1/0 = +\infty 1/0 = + ∞
Let H H H finite subset of A A A x 0 x_0 x 0 H H H f + g f + g f + g
f ( x 0 ) + g ( x 0 ) ≤ sup x ∈ H f ( x ) + sup x ∈ H g ( x ) ;
f(x_0) + g(x_0) \leq \sup_{x \in H} f(x) + \sup_{x \in H} g(x);
f ( x 0 ) + g ( x 0 ) ≤ x ∈ H sup f ( x ) + x ∈ H sup g ( x ) ; on the other hand, if
x 1 x_1 x 1 is one of the points of
H H H where
f f f takes its greatest value, then
f ( x 1 ) + g ( x 1 ) ≥ sup x ∈ H f ( x ) + inf x ∈ H g ( x ) ;
f(x_1) + g(x_1) \geq \sup_{x \in H} f(x) + \inf_{x \in H} g(x);
f ( x 1 ) + g ( x 1 ) ≥ x ∈ H sup f ( x ) + x ∈ H inf g ( x ) ; therefore
sup x ∈ H f ( x ) + inf x ∈ H g ( x ) ≤ sup x ∈ H ( f ( x ) + g ( x ) ) ≤ sup x ∈ H f ( x ) + sup x ∈ H g ( x ) .
\sup_{x \in H} f(x) + \inf_{x \in H} g(x) \leq \sup_{x \in H} (f(x) + g(x)) \leq \sup_{x \in H} f(x) + \sup_{x \in H} g(x).
x ∈ H sup f ( x ) + x ∈ H inf g ( x ) ≤ x ∈ H sup ( f ( x ) + g ( x )) ≤ x ∈ H sup f ( x ) + x ∈ H sup g ( x ) . The inequalities (17) and (18) follow from this by applying Proposition 5 of no. 4 and Theorem 1 of no. 2. The proofs of the other inequalities are analogous.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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