2
votes
3answers
90 views

If $I_{n+1}\subset I_n$, show that $\bigcap_{n=1}^\infty I_n$ is nonempty

Question: If $I_n$ is closed and bounded, $I_{n+1}\subset I_n$, and $I_n\neq\emptyset$, show that $\bigcap_{n=1}^\infty I_n$ is nonempty. This is not a homework help question. I'm actually looking ...
1
vote
3answers
68 views

Proving the open interval $(0,1)$ is uncountable [duplicate]

I am currently able to prove this statement using the Cantor diagonalisation argument, my question is whether there is another way (more simple or more complex) to prove this statement, without ...
3
votes
0answers
52 views

How to give an epsilon-delta proof of this limit statement? [duplicate]

Although I know a couple of proofs of the statement $$ \lim_{x \to 0 } \frac{\sin x}{x} = 1, $$ I would like to be able to come up with a proof using the definition of the limit (i.e. an ...
2
votes
3answers
741 views

Extreme Value Theorem proof help

Extreme Value Theorem: If $f$ is a continuous function on an interval [a,b], then $f$ attains its maximum and minimum values on [a,b]. Proof from my book: Since $f$ is continuous, then $f$ has the ...
6
votes
1answer
304 views

Proof that a certain entire function is a polynomial

Let $n\in\mathbf{N}$ be fixed, and $f$ entire and $|f^{-1}(\left\lbrace w\right\rbrace)|\leq n$ for every $w\in\mathbf{C}$. Then $f$ is a polynomial of degree at most $n$. I try to prove this ...
1
vote
1answer
286 views

Intuitive proof of Tucker's lemma or Borsuk-Ulam theorem

I'm looking for a intuitive proof of Tucker's lemma and/or the Borsuk-Ulam theorem. The proof should not make use of topology, cohomology etc. as it should be understandable by undergraduates. ...