# i have proved this, does it imply boundedness?

• Oct 19th 2008, 06:53 AM
szpengchao
i have proved this, does it imply boundedness?
f : X --> R

I proved: $\displaystyle \exists M\geq 0, \forall \epsilon>0, \forall x\in X, M-\epsilon < |f(x)|< M+\epsilon$
• Oct 19th 2008, 07:59 AM
ThePerfectHacker
Quote:

Originally Posted by szpengchao
f : X --> R

I proved: $\displaystyle \exists M\geq 0, \forall \epsilon>0, \forall x\in X, M-\epsilon < |f(x)|< M+\epsilon$

I think you actually probably proved $\displaystyle M - \epsilon < f(x) < M + \epsilon$ for $\displaystyle x\in X$.

Yes, this proves that $\displaystyle f$ is bounded on $\displaystyle X$.
• Oct 19th 2008, 09:26 AM
Plato
Quote:

Originally Posted by szpengchao
f : X --> R

I proved: $\displaystyle \exists M\geq 0, \forall \epsilon>0, \forall x\in X, M-\epsilon < |f(x)|< M+\epsilon$

If you can show that $\displaystyle \exists M\geq 0, \forall \epsilon>0, \forall x\in X, |f(x)|< M+\epsilon$
then it follows that $\displaystyle \forall x\in X, |f(x)|\le M$ or $\displaystyle f$ is bounded on $\displaystyle X$.