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

Lorena Becker

Lorena Becker

Answered question

2022-11-24

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

Answer & Explanation

Laylah Henry

Laylah Henry

Beginner2022-11-25Added 7 answers

The indermediate value theorem says:
Let f : [ a , b ] R be continuous and f ( a ) < 0 and f ( b ) > 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 = ξ
As 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 ) < 0 n
and
f ( b n ) > 0 n
Hence we know
lim n f ( a n ) 0
and
lim n f ( b n ) 0
Hence
0 f ( ξ ) 0
Hence f ( ξ ) = 0
You 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

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?