[Math] short proof for the Intermediate Value Theorem

analysiscalculusconnectednesscontinuity

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

Best Answer

The indermediate value theorem says:

Let $f:[a,b]\to \mathbb{R}$ be continuous and $f(a)<0$ and $f(b)>0$, then there exists a $\xi \in (a,b)$ such that $f(\xi)=0$.

You can prove it by using nested intervals:
You look at $f(\frac{a+b}{2})$, when it is bigger than null you look at $f$ on the interval $[a,\frac{a+b}{2}]$, if it is smaller than 0 we look at $[\frac{a+b}{2},b]$, 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 $(b-a)\cdot 2^{-n}$ which clearly converges to zero we have $$\lim_{n \to \infty} a_n =\lim_{n\to \infty} b_n=\xi$$ As $f$ is continuous we get $$\lim_{n\to \infty} f(a_n)=\lim_{n\to \infty} f(b_n)=f(\xi)$$ On the other hand we know $$f(a_n) < 0 \quad \forall n$$ and $$ f(b_n)>0 \quad \forall n $$ Hence we know $$\lim_{n\to \infty} f(a_n)\leq 0$$ and $$\lim_{n\to \infty} f(b_n)\geq 0$$ Hence $$0\leq f(\xi) \leq 0$$ Hence $f(\xi)=0$

You use that when $C_i$ is closed, bounded and non empty for all $i$ and $C_{i+1} \subset C_i$ for all $i$ then $$\bigcap_{i \in \mathbb{N}} C_i \neq \varnothing$$

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