The function g : ( 0 , ∞ ) → R , is continuous, g ( 1 ) &gt; 0 and lim x → ∞ g ( x ) = 0It is a fact that for every y between 0 and g ( 1 ) the function takes on a value in ( y , g ( 1 ) )How would one show that if: lim e → 0 + sup { g ( x ) : 0 &lt; x &lt; e } = 0then g attains a maximum value on 0 to infinity. Does g necessarily have to be bounded below for this to hold? Why?

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The function   g  :  (  0  ,  ∞  )  →      R  , is continuous,   g  (  1  )  &amp;gt;  0 and      lim          x      →      ∞        g  (  x  )  =  0It is a fact that for every   y between 0 and   g  (  1  ) the function takes on a value in   (     y  ,     g  (  1  )     )How would one show that if:      lim          e      →      0      +        sup  {  g  (  x  )  :  0  &amp;lt;  x  &amp;lt;  e  }  =  0then   g attains a maximum value on 0 to infinity. Does   g necessarily have to be bounded below for this to hold? Why?

ordonansexa · Accepted Answer

Hints1. Since       lim          x      →      ∞        g  (  x  )  =  0, there exists   R  ∈            R        +   such that for   x  &amp;gt;  R,   g  (  x  )  &amp;lt;  g  (  1  ).2. Since       lim          ϵ      →              0        +              sup  {  g  (  x  )  :  0  &amp;lt;  x  &amp;lt;  ϵ  }  =  0, there exists   r  ∈            R        +   such that for   x  ∈  (  0  ,  r  ),   g  (  x  )  &amp;lt;  g  (  1  ).3.   r  &amp;lt;  1  &amp;lt;  R.4. The set   [  r  ,  R  ] is a compact set, and a continuous function on a compact domain achieves an absolute maximum on that domain. This is the Extreme Value Theorem.

The function g:(0,infty)->R, is continuous, g(1)>0 and lim_(x->infty)g(x)=0 It is a fact that for every y between 0 and g(1) the function takes on a value in ( y, g(1) ) How would one show that if: lim_(e->0+)sup{g(x):0<x<e}=0

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