# Math Help - i have proved this, does it imply boundedness?

1. ## i have proved this, does it imply boundedness?

f : X --> R

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

2. Originally Posted by szpengchao
f : X --> R

I proved: $\exists M\geq 0, \forall \epsilon>0, \forall x\in X, M-\epsilon < |f(x)|< M+\epsilon$
I think you actually probably proved $M - \epsilon < f(x) < M + \epsilon$ for $x\in X$.

Yes, this proves that $f$ is bounded on $X$.

3. Originally Posted by szpengchao
f : X --> R

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