Say there is a sequence of points: {[tex] x_k,y_k [/tex]} that has a convergent subsequence:(adsbygoogle = window.adsbygoogle || []).push({});

{[tex] {x_k_i,y_k_i}} [/tex]} that converges to: [tex] (x_0,y_0) [/tex].

Sorry for poor latex, it should read "x sub k sub i"

Can I extrapolate the sequence {[tex]x_k_i[/tex]} and say it converges to [tex] x_0 [/tex] seperately?

The reason I ask this is because I would like to show that the projection of a compact set S in the x,y plane to the x-axis is also compact. Basically picking a sequence [tex] x_k [/tex] in the projection , finding a corresponding sequence {[tex] x_k,y_k [/tex]} in S where [tex] y_k [/tex] is arbitrary, that has a convergent subsequence whose limit is [tex] (x_0,y_0) [/tex], but then if I can bust that subsequence apart I can show the sequence in the projection has a convergent subsequence thus proving compactness (since sequentially compactness implies compactness for subsets of R^n)

Or do I need to project the subsequence in S down to the x-axis first? It seems like kind of "hand waving math" to just pull apart the subsequence and say each sequence of coordinates converges to a particular coordinate. Could someone point me in the right direction?

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# Sequences and Subsequences

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