What is the Difference Between Limit Superior and Limit Inferior in Sequences?

[kaosAD]

I have questions regarding this subject.

By definition, $$\limsup_{k \to \infty} f(x_k) \equiv \lim_{k \to \infty} \sup_{n \geq k} f(x_n)$$ and $$\liminf_{k \to \infty} f(x_k) \equiv \lim_{k \to \infty} \inf_{n \geq k} f(x_n)$$. Say a sequence $$\{x_k\}$$ converging to $$0$$ from the left in the following example.

$$f(y) = \left\{ <br /> \begin{array}{ll}<br /> y + 1 & \quad ,y > 0 \\<br /> y & \quad ,y \leq 0<br /> \end{array}<br /> \right.$$

Then $$\limsup_{k \to \infty} f(x_k) = \liminf_{k \to \infty} f(x_k) = f(0)$$.

Suppose we have another sequence $$\{x_k\}$$ converging to $$0$$ from the right. Then $$\limsup_{k \to \infty} f(x_k) = \liminf_{k \to \infty} f(x_k) > f(0)$$.

What is the difference between $$\limsup_{k \to \infty} f(x_k)$$ and $$\liminf_{k \to \infty} f(x_k)$$? I don't see any difference.

 

[Hurkyl]

I'm very confused as to what you're actually doing.

Anyways, your function has an actual limit at $+\infty$: in particular,

$$\lim_{y \rightarrow +\infty} f(y) = +\infty$$

so of course the lim sup and lim inf are going to be equal.


I suspect you meant to take a one sided limit at zero.. but again, the function has an actual limit there:

$$\lim_{y \rightarrow 0^+} f(y) = 1$$

so again, the lim sup and lim inf are going to be equal.



Oh, I think I've figured out what you're trying to do.

What you're forgetting is that

$$\limsup_{y \rightarrow 0} f(y)$$

is the supremum over all sequences that converge to zero and for which $f(x_k)$ converges. So just picking a particular sequence and saying that the lim sup equals the limit of that sequence is wrong.

 

[kaosAD]

I'm very confused as to what you're actually doing.

Sorry for not explaining my problem clearer.

Oh, I think I've figured out what you're trying to do.

What you're forgetting is that

$$\limsup_{y \rightarrow 0} f(y)$$

is the supremum over all sequences that converge to zero and for which $f(x_k)$ converges. So just picking a particular sequence and saying that the lim sup equals the limit of that sequence is wrong.

That was what my problem was. Many thanks, Hurkyl! :)
Does this mean $\lim_{k \to \infty} f(x_k)$ is the limit over all sequences as well?

Also, using the given example, can I write $0 \leq \lim_{x \to 0} f(x) \leq 1$? or that $\lim_{x \to 0} f(x)$ does not exist, so we cannot write down its range?

 

[Hurkyl]

You can't write that expression for the limit, because as you said, the limit is undefined.

You can say that all of the limit points are in that interval, though, because lim inf and lim sup are the minimum and maximum of the limit points.