My final for my introductory analysis course is tomorrow and my teacher gave us a list of possible theorems to prove. If anyone could please show me a proof for The Intermediate Value Theorem that is short and easy to follow, so even if I still cannot understand it I can at least memorize it. Also, I have looked through numerous texts and the internet, but they all seem to confuse me. I know that itis an insult to all you math experts to memorize proofs, but I am desperate at this point. Thank you

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Laylah Henry · Accepted Answer

The indermediate value theorem says:Let   f  :  [  a  ,  b  ]  →      R   be continuous and   f  (  a  )  &amp;lt;  0 and   f  (  b  )  &amp;gt;  0, then there exists a   ξ  ∈  (  a  ,  b  ) such that   f  (  ξ  )  =  0.You can prove it by using nested intervals:You look at   f  (            a      +      b        2    ), when it is bigger than null you look at   f on the interval   [  a  ,            a      +      b        2    ], if it is smaller than 0 we look at   [            a      +      b        2    ,  b  ], when it is 0 we are done. Lets denote the left endpoints with       a    n   and the right endpoints with       b    n  .As the diameter of our nested intervals is   (  b  −  a  )  ⋅      2          −      n       which clearly converges to zero we have      lim          n      →      ∞            a    n    =      lim          n      →      ∞            b    n    =  ξAs   f is continuous we get      lim          n      →      ∞        f  (      a    n    )  =      lim          n      →      ∞        f  (      b    n    )  =  f  (  ξ  )On the other hand we know  f  (      a    n    )  &amp;lt;  0    ∀  nand  f  (      b    n    )  &amp;gt;  0    ∀  nHence we know      lim          n      →      ∞        f  (      a    n    )  ≤  0and      lim          n      →      ∞        f  (      b    n    )  ≥  0Hence  0  ≤  f  (  ξ  )  ≤  0Hence   f  (  ξ  )  =  0You use that when       C    i   is closed, bounded and non empty for all   i and       C          i      +      1        ⊂      C    i   for all   i then      ⋂          i      ∈              N                  C    i    ≠  ∅

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