Can someone help me understand the proof for this theorem:
All bounded sequences in $R^n$ have a sub-sequence that is convergent
Here's the proof:
$Suppose \: n=2 \:and\: u_{j}=\binom{x_{j}}{y_{j}} \\ We\:suppose\:that\:u_{j}\:is\:bounded\:for\:||\cdot ||_{\infty} \:Which\:means: \\ \exists M\geq 0, \forall j,\:max(|x_{j}|,|y_{j}|)\geq M(so\: (x_{j})\:and\:(y_{j})\:are\:bounded)\\ 1)(x_{j})\:is\:bounded:\\ \exists\:subsequence \:\tau_{j}(that\:is\:strictly\:increasing)\\ 2)The\:sequence\:(y_{\sigma_{j} })\:is\:bounded,\:so\:it\:has\:a\:subsequence.\:The\:associated\:subsequence\:to\:(x_{\tau_{j}})\:converges$
I am having a difficult time to understand the second part. The $\delta$ and $\sigma$ sequences just seem to have no connection to me.