Limit Superior/Inferior

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.

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

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.

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Hurkyl
Staff Emeritus
Gold Member
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.

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Hurkyl said:
I'm very confused as to what you're actually doing.
Sorry for not explaining my problem clearer.

Hurkyl said:
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
Staff Emeritus