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How can i prove that a limit of a convex functions is convex. (a convergent sequence)

I knew that, If a sequence of convex continuous functions defined on a compact subset D of Rⁿ is convergent, then the limit function is convex on D. I do not know how to show this at all.

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Hint: If $f_n\to f$ pointwise then $$f_n(\lambda a+(1-\lambda) b)\to f(\lambda a+(1-\lambda) b)$$ and $$\lambda f_n(a)+(1-\lambda)f_n(b)\to \lambda f(a)+(1-\lambda)f(b).$$

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exactly the thing I wanted :) –  Lost1 May 17 '13 at 12:09
    
perhaps also indicates my laziness trying to prove something myself. –  Lost1 May 17 '13 at 12:10
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