Let : [math]f_{n}(x) =\frac{x}{x^2+n}[/math] be a sequence of functions in the Real Nos.

 

I can prove point wise convergence to the zero function as n goes to infinity .

 

But is there a uniform convergence ??

 

If there is, can anyone prove it ,please??

[math]N=\bigl{\lceil}1/(4\epsilon^2)\bigr{\rceil}[/math]

[math]N=\bigl{\lceil}1/(4\epsilon^2)\bigr{\rceil}[/math]

 

Do you know any natural No N to be equal to [math] \frac{1}{4\epsilon^2}[/math]

Learn your symbols, triclino. [math]\lceil x\rceil[/math] is the the ceiling function. For example, [math]\lceil \pi \rceil = 4[/math].

 

Edit

I should have said [math]N=\lceil 1/(4\epsilon^2)\rceil+1[/math] to handle the rare case where [math]1/(4\epsilon^2)[/math] is an integer.

Learn your symbols, triclino. [math]\lceil x\rceil[/math] is the the ceiling function. For example, [math]\lceil \pi \rceil = 4[/math].

 

Edit

I should have said [math]N=\lceil 1/(4\epsilon^2)\rceil+1[/math] to handle the rare case where [math]1/(4\epsilon^2)[/math] is an integer.

 

You mixing up ceiling function with the floor function.

 

But according to what axiom or theorem you came to the conclusion :

 

[math]N=\lceil 1/(4\epsilon^2)\rceil+1[/math] or [math]N=\lceil 1/(4\epsilon^2)\rceil[/math]

You mixing up ceiling function with the floor function.

 

No, this is standard notation. The floor function is [math]\lfloor x\rfloor[/math].

 

But according to what axiom or theorem you came to the conclusion :

 

[math]N=\lceil 1/(4\epsilon^2)\rceil+1[/math] or [math]N=\lceil 1/(4\epsilon^2)\rceil[/math]

 

Plug it into the definition of uniform continuity and check that the definition is satisfied.

For crying out loud, triclino. Do look things up first. Here: http://en.wikipedia.org/wiki/Floor_and_ceiling_functions.

 

What is the definition of uniform convergence? What is the maximum of the absolute value of your function less the limit of the sequence?

No, this is standard notation. The floor function is [math]\lfloor x\rfloor[/math].

 

 

 

Plug it into the definition of uniform continuity and check that the definition is satisfied.

 

You mixing up uniform continuity with uniform convergence .

 

But in my very 1st post i ask if any body could prove uniform convergence that i could not prove.

 

Now by producing a No and plugging it into the definition of uniform convergence to find out if the definition is satisfied or not ,i am sorry is not much of a help

Triclino, instead of arguing (and arguing with an administrator is not very bright), try the hint. Heck, it wasn't a hint. I gave you the answer.