Definition of liminf of sequence of functions?

[mathmonkey]

Homework Statement

Hi I've come across the term lim inf $f_n$ in my text but am not sure what it means.

$\lim \inf f_n = \sup _n \inf _{k \geq n} f_k$

In fact, I am not sure what is supposed to be the output of lim inf f? That is, is it supposed to return a real-valued number, or a function itself?

Generally for a real-valued function $\inf f$ refers to $\inf_x f(x)$. That is, it returns the largest real-valued number smaller than f(x) for all x. If that's the case, then it should follow that $\lim \inf f_n$ also returns a real-valued number? But I've always thought the implication of lim inf and lim sup is that if $f_n$ converges uniformly to $f$ then

$\lim \inf f_n = \lim \sup f_n = \lim f_n = f$

but that doesn't seem to hold if i use this definition? Any help or clarification would be greatly appreciated. Thanks!

 

[Mandelbroth]

Homework Statement

Hi I've come across the term lim inf $f_n$ in my text but am not sure what it means.

I really like this image, from Wikipedia. It helped get me started in understanding intuitively what limsup and liminf mean. Hopefully it does the same for you.

 

[mathmonkey]

Hi Mandelbroth,

Thanks for your reply! However, that picture only describes the limsup/liminf of a sequence of points, which is itself a point, which is easier to intuit for me. But what I'm wondering is what is the limsup/liminf of a sequence of functions supposed to be? Is it to be a function itself? Or just a point? I am unclear as to the definition itself of limsup for sequences of functions...

So far I'm thinking it shouldn't be intuitive for limsup/liminf f_n to be a point, since as I mentioned earlier if $f_n$ converges to $f$ uniformly it ought to be the case that $\lim \sup f_n = \lim \inf f_n = \lim f_n$? Thanks again for any help

 

[Dick]

Hi Mandelbroth,

Thanks for your reply! However, that picture only describes the limsup/liminf of a sequence of points, which is itself a point, which is easier to intuit for me. But what I'm wondering is what is the limsup/liminf of a sequence of functions supposed to be? Is it to be a function itself? Or just a point? I am unclear as to the definition itself of limsup for sequences of functions...

So far I'm thinking it shouldn't be intuitive for limsup/liminf f_n to be a point, since as I mentioned earlier if $f_n$ converges to $f$ uniformly it ought to be the case that $\lim \sup f_n = \lim \inf f_n = \lim f_n$? Thanks again for any help

Yes, it's a function. The value of lim inf $f_n$ at a point x is the lim inf of sequence of numbers $f_n(x)$ as n->infinity.

 

[micromass]

Hi Mandelbroth,

Thanks for your reply! However, that picture only describes the limsup/liminf of a sequence of points, which is itself a point, which is easier to intuit for me. But what I'm wondering is what is the limsup/liminf of a sequence of functions supposed to be? Is it to be a function itself? Or just a point? I am unclear as to the definition itself of limsup for sequences of functions...

If you understand limsup/liminf of sequences of points, then this isn't too hard. Basically, you take a sequence of functions $(f_n)_n$. Now, if I take a fixed $x$, then $x_n = f_n(x)$ is a sequence of points. So the liminf makes sense. Now, we define

$$f(x) = \liminf x_n$$

And we do that for any point. So the liminf of a sequence of functions is again a function $f$ which satisfies that

$$f(x) = \liminf f_n(x)$$

for any $x$. So you evaluate the liminf pointswize.

So far I'm thinking it shouldn't be intuitive for limsup/liminf f_n to be a point, since as I mentioned earlier if $f_n$ converges to $f$ uniformly it ought to be the case that $\lim \sup f_n = \lim \inf f_n = \lim f_n$? Thanks again for any help

This is true. But uniform convergence isn't even needed. Pointswize convergence is enough.

 

[mathmonkey]

Thanks guys! That makes perfect sense!