Homework Help: Lim sup, lim inf definition/convention

1. May 11, 2007

quasar987

My book says that if the set of all cluster points is empty, then we write lim sup = $-\infty$ and if the sequence is not bounded above, we write limsup = $+\infty$.

But what if both happen at the same time? for instance consider x_n=1/n. There are no accumulation points and it is unbounded above.

2. May 11, 2007

NateTG

Maybe I'm getting too tired, but
$$1>\frac{1}{n}$$
is an upper bound, and
$$0$$
is an accumulation point.

You might consider $x_n=n(-1)^n$ which is unbounded in the reals, and doesn't have any real accumulation points. However, in the extedended reals, $\pm \infty$ are cluster points of that sequence, and $+ \infty$ is an upper bound for all sequences.

3. May 11, 2007

quasar987

It is I who is too tired. I meant to say x_n = n.

4. May 11, 2007

HallsofIvy

I don't see a problem. Since the set of all cluster points is empty, lim sup= $-\infty$ and since the sequence is not bounded above, limsup= $\infty$.

5. May 11, 2007

quasar987

Well, actually, the author give the same x_n=n as an example and he write lim sup=+$\infty$.

As if the fact that it is not bounded above takes priority over the fact that the set of all cluster points is empty.

6. May 11, 2007

AKG

Are "lim sup" and "limsup" supposed to be two different things?

7. May 11, 2007

No, no.

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