Suppose that f : [ a , b ] → R is continuous and that f ( a ) &lt; 0 and f ( b ) &gt; 0. By the intermediate-value theorem, the set S = { x ∈ [ a , b ] : f ( x ) = 0 } is nonempty. If c = sup S, prove that c ∈ S.My first thought was to show that S is finite therefore c ∈ S, but there is no guarantee that f doesn't have infinitely many zeros.A thought I have now is to show that c &gt; max ( S ) can not be true, but I do not know how to show this, is this even the correct thing to show? Thank you in advance for any input.

Question

Suppose that   f  :  [  a  ,  b  ]  →      R   is continuous and that   f  (  a  )  &amp;lt;  0 and   f  (  b  )  &amp;gt;  0. By the intermediate-value theorem, the set   S  =  {  x  ∈  [  a  ,  b  ]  :  f  (  x  )  =  0  } is nonempty. If   c  =  sup  S, prove that   c  ∈  S.My first thought was to show that   S is finite therefore   c  ∈  S, but there is no guarantee that   f doesn&#039;t have infinitely many zeros.A thought I have now is to show that   c  &amp;gt;  max  (  S  ) can not be true, but I do not know how to show this, is this even the correct thing to show? Thank you in advance for any input.

Nollothwnfcm · Accepted Answer

You&#039;ve noted that when   S is finite, then   c  =  sup  S  ∈  S  .Now what if   S is infinite? Then there exists a sequence in   S that converges to   sup  S  .. (There are several ways to prove this. Try it on your own if you don&#039;t already know how. But a standard proof would be to choose       c    0  , then choose       c    1   such that it&#039;s bigger than       c    0   but less than   c. We know such a number must exist because if not,       c    0   would be a smaller upper bound than   c. And we continue on in this fashion.) Call this sequence             c      n        . Now as   f is continuous,  f  (  c  )  =      lim          n      →      ∞        f  (      c    n    )  .Thus   f  (  c  )  =  0.

Suppose that f : [ a , b ] → <mrow class="MJX-TeXAtom-O

Answered question

Answer & Explanation

New Questions in Calculus 2