Consider this image on Wikipedia that attempts to graphically represent the concepts of $\underset{n \rightarrow \infty}{\text{lim sup}} \space a_n$ and $\underset{n \rightarrow \infty}{\text{lim inf}} \space a_n$.

Now, I realize that there is some sequence $ (x_n)_{n \in \mathbb{N}}$, whose graph for $n \leq 600$ is identical to the one shown on the photo, such that the dashed black lines represent the limit superior and limit inferior of that sequence.

But what if I define $x_n := 0$ for every $n > 600$? Wouldn't the sequence, then, be convergent and make the current depiction of limit superior and limit inferior (as shown on the photo) wrong?

In other words, it seems that when one only has this photo, then one cannot determine precisely what the limit superior and limit inferior of the sequence are (but only an upper bound and a lower bound for those limits), because one wouldn't know how the sequence is defined for large values of $n$.

Am I getting something wrong?

  • 2
    $\begingroup$ You aren't wrong. The picture comes with the implicit assumption that the sequence continues to "look like that" for larger values of $n$. We cannot plot infinitely many values after all. $\endgroup$ – Thorgott Nov 17 '19 at 14:28

If $(x_n)$ is a sequence and only the values of $x_n$ for $n\leq m$ are known where $m$ is some fixed positive integer then nothing is known about $\limsup x_n$ and $\liminf x_n$.

This is illustrated by the fact that for the sequence $(y_n)_n$ prescribed by $y_n=x_{n+m}$ we have:$$\limsup y_n=\limsup x_n\text{ and }\liminf y_n=\liminf x_n$$

We could say that $\limsup$ and $\liminf$ are determined by the "tail" of the sequence.

  • $\begingroup$ But if we do additionally know $ \underset{n \geq m}{\text{sup}} \space x_n$ and $\underset{n \geq m}{\text{inf}} \space x_n$, then it seems we would have an upper bound and a lower bound, respectively, for the limit superior and limit inferior of the sequence. Is that correct? $\endgroup$ – ðulfiqaːr Nov 17 '19 at 14:47
  • $\begingroup$ Yes that is correct. $\endgroup$ – drhab Nov 17 '19 at 16:25

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