MHB Can a sequence without a convergent subsequence have a limit of infinity?

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A sequence of positive real numbers without a convergent subsequence must be unbounded, leading to the conclusion that its limit as n approaches infinity is positive infinity. However, if the sequence includes both positive and negative values, the absolute values of the terms must also approach infinity. The discussion highlights that being unbounded does not guarantee that the limit is infinity, as demonstrated by the sequence x_n = |tan(n)|, which oscillates and does not converge. The clarification emphasizes that for a sequence to have a limit of infinity, it must not have a bounded subsequence. Thus, the relationship between boundedness and convergence is crucial in determining the limit behavior of sequences.
Fernando Revilla
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I quote a question from Yahoo! Answers

Let (xn)n be a sequence of positive real numbers that has no convergent subsequence. Prove that lim(n→∞) x of n=+∞. What if the xn are permitted to take both positive and negative values?

I have given a link to the topic there so the OP can see my response.
 
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Each bounded sequence has a convergent subsequence, so $x_n$ has no convergent subsequence implies that $x_n$ is not bounded. In this case, $x_n\to +\infty$ if $x_n$ is a sequence of positive real numbers and $\left|x_n\right|\to +\infty$ if the $x_n$ are permitted to take both positive and negative values.
 
Fernando Revilla said:
Each bounded sequence has a convergent subsequence, so $x_n$ has no convergent subsequence implies that $x_n$ is not bounded. In this case, $x_n\to +\infty$ if $x_n$ is a sequence of positive real numbers and $\left|x_n\right|\to +\infty$...

The fact that the positive term sequence $x_{n}$ is unbounded doesn't mean that $\displaystyle \lim_{n \rightarrow \infty} x_{n} = + \infty$. As example You can consider the sequence $\displaystyle x_{n} = |\tan n|,\ n \ge 1$...

Kind regards

$\chi$ $\sigma$
 
chisigma said:
The fact that the positive term sequence $x_{n}$ is unbounded doesn't mean that $\displaystyle \lim_{n \rightarrow \infty} x_{n} = + \infty$. As example You can consider the sequence $\displaystyle x_{n} = |\tan n|,\ n \ge 1$...

Right, my fault, I was thinking about a monotic sequence.
 
The negation of $\lim_{n\to\infty}x_n=\infty$ says that $x_n$ has a bounded subsequence and therefore a convergent sub-subsequence.
 
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