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Let $x_n$ be a sequence of real numbers.

Definition: $x \in \mathbb R \cup \{-\infty,\infty\}$ is a limit point of a sequence $x_n$ if there is a subsequence $x_{n_k}$ of our sequence such that $x_{n_k} \to x$.

If $A= \{x \in \mathbb R \cup \{-\infty,\infty\} \mid x \text { is a limit point of }x_n\}$, prove that $A$ is not empty.

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This is "essentially" (but not the same as) the Bolzano-Weierstrass theorem. Do you know this theorem? What results are you allowed to use? –  Srivatsan Dec 1 '11 at 19:08
Have you considered proving $\mathbb{R} \cup \{ - \infty, + \infty\}$ is compact? –  Isaac Solomon Dec 1 '11 at 19:10
Isn't Bolzano-Weierstrass theorem for bounded sequences? Well, I am allowed to use pretty much everything I know. I am only starting to learn topology. –  Mihai Bogdan Dec 1 '11 at 19:12
Could you please expand that ideea, Isaac S.? I don't really understand that notion. –  Mihai Bogdan Dec 1 '11 at 19:16
@MihaiBogdan So, if Bolzano-Weierstraß works for bounded sequences, maybe you can treat unbounded sequences separately. –  Phira Dec 1 '11 at 19:18
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4 Answers

up vote 7 down vote accepted

If the sequence is bounded, then you can invoke Bolzano-Weierstrass theorem.

So we can assume it is unbounded. Let us assume it is not bounded from above. (The symmetric case is similar.)

Thus for every $C$ there exists $n$ such that $x_n>C$.

Start with $C_1=1$ and choose $n_1$ such that $x_{n_1}>C_1=1$.

Now choose $C_2=2+\max\{x_j\; : \; j\le n_1\}$. Obviously $C_2\ge 2$ and you can choose $n_2$ such that $n_2>n_1$ and $x_{n_2}>C_2$.

You continue by induction. In the $k$-th step you choose $C_k=k+\max\{x_j \; : \; j\le n_k\}$. This means that $C_k\ge k$ and you there exists $n_{k+1}>n_k$ with $x_{n_k}>C_k$.

In this way you obtain a subsequence $n_k$ such that $x_{n_k} \ge k$. This implies that this subsequence converges to $+\infty$.

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Thank you Martin Sleziak, I understand this proof. Thanks, everyone. –  Mihai Bogdan Dec 1 '11 at 19:28
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A proof sketch assuming Bolzano-Weierstrass.

  1. Show that $x_n$ is bounded above if and only if $\infty$ is not a limit point of $x_n$.

  2. Reversing the above argument, show that $x_n$ is bounded from below if and only if $- \infty$ is not a limit point.

  3. From (1.) and (2.), conclude that if neither $\infty$ nor $-\infty$ is a limit point of $x_n$, then $x_n$ is bounded. Further, using Bolzano-Weierstrass, conclude that if neither $\infty$ nor $-\infty$ is a limit point of $x_n$, then $x_n$ has a subsequence converging to some limit $x \in \mathbb R$.

To summarize the argument, the key idea is that the sequence is

  • either unbounded, in which case one of $+\infty$ and $-\infty$ is a limit point;
  • or it's bounded, in which case Bolzano-Weierstrass shows the existence of a limit point in $\mathbb R$.

Note: Martin's answer explains the idea behind (1.). :-)

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$\arctan : [-\infty,\infty] \rightarrow [-\pi/2,\pi/2]$. The sequence $\{x_n\}_{n=1}^\infty \subseteq \mathbb{R}$ has a limit point in $[-\infty,\infty]$ if and only if the sequence $\{\arctan x_n\}_{n=1}^\infty \subseteq [-\pi/2,\pi/2]$ has a limit point in $[-\pi/2,\pi/2]$. Bolzano-Weierstrass takes care of that.

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Define $y_n = \sup \{ x_n, x_{n+1}, x_{n+2}, \cdots \} .$ Then $y_n$ is a monotone decreasing sequence and hence $ Y = \displaystyle \lim_{n\to\infty} y_n$ exists (possibly $-\infty$).

The claim is that $ Y \in A .$ If $Y$ is finite, then argue as follow: For any $\epsilon > 0 $, by the definition of supremum for each $n\in \mathbb{N}$ we can find $ n_k \geq n $ such that $ |y_n - x_{n_k} | < \frac{\epsilon}{2}.$ By definition of limit, for some $n_0 \in\mathbb{N}$ we have $ |Y - y_n| < \frac{\epsilon}{2} $ for all $ n > n_0.$ Thus for all $n> n_0$, $$ |Y - x_{n_k}| \leq |Y-y_n| + |y_n - x_{n_k}| < \epsilon.$$ Thus, $ x_{n_k} \to Y.$

I leave the case where $ Y= -\infty$ to you.

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Note that it is possible that all $y_n$'s are equal to $+\infty$. (E.g. the sequence $x_n=n$.) So perhaps you should deal with the case $Y=\infty$ too. (Or at least you should write that this case is also left to the reader.) –  Martin Sleziak Dec 1 '11 at 20:05
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