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Subsequence converging

1. Homework Statement

Consider the sequence [tex]\left\{ x_{n} \right\}[/tex].

Then [tex] x_{n}[/tex] is convergent and [tex]\lim x_{n}=a[/tex] if and only if, for every non-trivial convergent subsequence, [tex]x_{n_{i}}[/tex], of [tex]x_{n}[/tex], [tex]\lim x_{n_{i}}=a[/tex].



2. Homework Equations
The definition of the limit of a series:
[tex]\lim {x_{n}} = a \Leftrightarrow [/tex] for every [tex]\epsilon > 0[/tex], there exists [tex] N \in \mathbb{N}[/tex] such that for every [tex]n>N[/tex], [tex]\left| x_{n} - a \right| < \epsilon[/tex].



3. The Attempt at a Solution

Ok, so I easily see how to show that it [tex]\lim {x_{n}} = a [/tex], then every convergent subsequence must also converge to [tex]a[/tex].
But I'm stuck on how to show the other way.
 

Answers and Replies

Dick
Science Advisor
Homework Helper
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I would say, well isn't a_n a subsequence of itself? But you also said 'non-trivial'. I'm not sure exactly what that means, but can't you split a_n into two 'non-trivial' subsequences, which then converge, but when put together make all of a_n?
 
Yeah, I asked my prof about this one. To him, apparently "non-trivial" just means that the subsequence is not equal to the original sequence. I don't think it actually has any bearing on the problem.

The trick, he said, is that you have to consider every non-trivial (convergent) subsequence.

I'm not sure I know what that means.
 
Dick
Science Advisor
Homework Helper
26,258
618
Ok, then suppose a_n has two convergent subsequences with different limits. Then does a_n have a limit?
 

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