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how can be prove that $\max(f(n),g(n)) = \theta(f(n)+g(n))$

How to prove $max(f(n),g(n)) = Θ(f(n)+g(n))$?

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marked as duplicate by J. M., JavaMan, anon, Zev Chonoles Sep 8 '11 at 20:43

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Is $\Theta(f(x))$ the same as $O(f(x))$ of "big-O" fame? – robjohn Sep 8 '11 at 19:08
@robjohn: $f = \Theta(g)$ if both $f = O(g)$ and $g = O(f)$. – JavaMan Sep 8 '11 at 19:16
up vote 3 down vote accepted

Let $h=\max(f,g)$. Then $f+g\le 2h$ and so $f+g=O(h)$. If $f$ and $g$ are positive, then $h \le f+g$ and so $h=O(f+g)$.

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