Thread: how to prove this..?

1. how to prove this..?

D is subset of real number in p dimension
[ $(\vec{x_{n}})$, n belongs natural number] is a subset of D
$\vec{x}$ belongs to D
suppose $(\vec{x_{n}})$, n belongs natural number converges to $\vec{x}$
f: D->R is cont' on D and f( $(\vec{x_{n}})$)<= r for all n belongs to natural number
Prove that f( $\vec{x}$)<=r

i have totally no idea about wt is happening here.
suppose $(\vec{x_{n}})$, n belongs natural number converges to $\vec{x}$
this sentence means lim n->infinity $(\vec{x_{n}})$ = $\vec{x}$?
and then that means lim n->infinity f( $\vec{x_{n}}$) = f( $\vec{x}$) ?
how can these help me to derive that...thank you so much

2. Originally Posted by pokemon1111
D is subset of real number in p dimension
[ $(\vec{x_{n}})$, n belongs natural number] is a subset of D
$\vec{x}$ belongs to D
suppose $(\vec{x_{n}})$, n belongs natural number converges to $\vec{x}$
f: D->R is cont' on D and f( $(\vec{x_{n}})$)<= r for all n belongs to natural number
Prove that f( $\vec{x}$)<=r

i have totally no idea about wt is happening here.
suppose $(\vec{x_{n}})$, n belongs natural number converges to $\vec{x}$
this sentence means lim n->infinity $(\vec{x_{n}})$ = $\vec{x}$?
and then that means lim n->infinity f( $\vec{x_{n}}$) = f( $\vec{x}$) ?
how can these help me to derive that...thank you so much
Have you tired a simple epsilon argument ?

Bobak

3. Originally Posted by bobak
Have you tired a simple epsilon argument ?

Bobak
what does it mean?

4. Originally Posted by pokemon1111
what does it mean?
An argument that uses the definition of limits and continuity.

Bobak

5. Originally Posted by bobak
An argument that uses the definition of limits and continuity.

Bobak
|| $(\vec{x_{n}})$- $\vec{x}$||< epsilon
implies
||f( $(\vec{x_{n}})$) - f( $\vec{x}$)||< delta
?