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Let $f,g:X\to\Bbb R$ be two functions where $X$ is any set. Then $$ \left|\sup_x f(x) - \sup_x g(x)\right|\leq \sup_x|f(x) - g(x)|. $$

This fact is fairly easy to prove, but it seems to be a rather well-known trick, so I am looking for a reference that contains it.

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As I know you speak Russian, there is a book Васин, Морозов "Теория игр и модели математической экономики". Lemma 2.2., page 23 –  Nimza Apr 2 '13 at 8:30
    
$||f||=\sup_{x\in X}|f(x)|$ is a norm (if f is finite). –  user59671 Apr 2 '13 at 8:31
    
@Nimza: thanks, perhaps I should have specified that ideally it is a reference in English :) –  Ilya Apr 2 '13 at 8:31
    
@CutieKrait: I think, rather $\|f\| = \sup_x|f(x)|$ is a norm, the one you wrote can take negative values. –  Ilya Apr 2 '13 at 8:32
    
yep. correcte‌d –  user59671 Apr 2 '13 at 8:33
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I have found that this fact appears as A.3 Proposition in the book "Adaptive Markov Control Processes" by O. Hernandez-Lerma, which in turn cites "Foundations of Non-stationary Dynamic Programming with Discrete Time Parameter" by K. Hinderer.

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