Let X ⊂ R be convex, and let f : X → R be concave. Prove that for every x ∈ X and every sequence { x n } converging to x ∈ X, we have lim n → ∞ inf f ( x n ) ≥ f ( x ) .

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Let   X  ⊂      R   be convex, and let   f  :  X  →      R   be concave. Prove that for every   x  ∈  X and every sequence   {      x    n    } converging to   x  ∈  X, we have      lim          n      →      ∞        inf  f  (      x    n    )  ≥  f  (  x  )  .

meexeniexia17h · Accepted Answer

We can split   (      x    n    ) into two sequences one of which stays above   x and another which stays below   x. It is enough to prove the inequality for each of these subsequences. Let   x  &amp;lt;      x    n    &amp;lt;  x  +  1 Then       x    n    =      t    n    x  +  (  1  −      t    n    )  (  x  +  1  ) where       t    n    =  x  +  1  −      x    n    →  1. Note that   0  &amp;lt;      t    n    &amp;lt;  1. Hence   f  (      x    n    )  ≥      t    n    f  (  x  )  +  (  1  −      t    n    )  f  (  x  +  1  ) and   lim  inf  f  (      x    n    )  ≥  (  1  )  (  f  (  x  )  )  +  (  0  )  (  f  (  x  +  1  )  )  =  f  (  x  ).There is a similar argument when   x  −  1  &amp;lt;      x    n    &amp;lt;  x

Let X sub R be convex, and let f:X->R be concave. Prove that for every x in X and every sequence {xn} converging to x in X, we have lim_(n->oo) inff(x_n)>=f(x).

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