- #1

Doom of Doom

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## 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].

## 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].

## 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.