How do you define sequences that converge to infinity?
For instance consider the sequence $\{1,0,2,0,3,0,4,0,..\}$ Intuitively we know that the sequence converges to $\infty$ but how do we check that rigorously. If I imitate the formal definition of convergence then I believe that we can at best come up with something like this:
$(x_n)\to\infty$ if for any $\epsilon>0$ there exists $N\in\mathbb{N}$ such that for $n\geq N$ we have $x_n>\epsilon.$
Now this definiton does help us in proving the convergence of some sequences such as $x_n=\sqrt{n}$ because in this case we can let $N\geq \epsilon^2.$ However this definition fails to show that the aforementioned sequence $\{1,0,2,0,3,0,4,0,..\}$ converges to $\infty$. I am thus guessing that "there exists" a better definition out there. So please suggest me some references or maybe provide me with a definition that is able to take care of convergence to $\infty$ in general.
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$\begingroup$The general idea behind limits converging to $\infty$ are:
- There is a topological notion of limit in terms of open sets or open neighborhoods rather than in terms of distance
- $+\infty$ and $-\infty$ are best understood as points in the extended real numbers
The definition you cited is, in fact, equivalent to what it means for a real-valued sequence to converge to $+\infty$ in the extended real numbers.
The problem is that the sequence you consider doesn't converge. In the extended real numbers, it is a divergent sequence with two limit points: $0$ and $+\infty$. In that regard, it's comparable to the sequence $0,1,0,1,0,1,\ldots$.
Without more examples or attempts at elaboration, I'm not sure what idea you have that you have incorrectly given the name "converging to $\infty$". One possibility is that you simply have in mind the idea of a sequence being unbounded above. This is a sequence having $+\infty$ as a limit point, or alternatively, something satisfying the property
$\endgroup$ $\begingroup$For all real $M$, there exists some $n$ such that $x_n > M$
The usual definition is that for every real number $x$ there is a positive integer $N$ such that $a_m>x$ for all $m\geq N$.
It is very similar to the usual definition of convergence, instead that we ask that all numbers be sufficiently "big" after a point. (instead of asking all numbers be sufficiently "close" to the limit after a point).
$\endgroup$ 1 $\begingroup$We say that the sequence $(x_n)$ diverges to infinity. This is therefore different from a sequence like $\left((-1)^n\right)$, which diverges but not to infinity.
The formal definition you give for divergence to infinity is essentially correct.
$\endgroup$ 2 $\begingroup$Your definition is fine (even if "converge to infinity" is a sub-optimal choice of words since convergence to a finite limit implies many useful properties that tending to infinity does not), it is the sequence that does not converge. If a sequence converges (or tends to infinity), then every subsequence should converge to the same limit (or tend to infinity). This does not hold for the subsequence $\{0,0,0,...\}$ (which does not tend to infinity).
The limit superior $\left(\limsup_{n \rightarrow \infty} x_n = \lim_{n \rightarrow \infty} \sup_{m \ge n} x_n\right)$ is infinite, but that is not sufficient for a claim of "tends to infinity". As Will R pointed out in a comment, the sequence $\{ 1,1,1,{1\over 2},1,{1\over 3},1,{1\over 4},...\}$ does not converge to zero despite its limit inferior being zero. Another (necessary but insufficient for a claim of "tends to infinity") property of the sequence is that it is not bounded from above $\left( \forall M \ \exists n \mid x_n \gt M \right)$ (as mentioned by Hurkyl)
A possibly useful (although not perfect) way of thinking about the definition of tending to infinity $\left( \lim_{n \rightarrow \infty} x_n = \infty \Leftrightarrow \forall k \ \exists N \mid n \gt N \Rightarrow x_n \gt k\right)$ is "every subsequence exceeds all bounds".
Sources: currently studying Computer Science
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