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 Quote by Office_Shredder If any subsequence of xn converges to a, then xn converges to a too. Rather, what you can conclude is a single subsequence of xn converges to a (which is what sequential compactness is about)
This is not true- consider a_n=(-1)^n=1,-1,1,-1,1,-1,...

Then a_{2n}=1 is a subsequence that converges to 1, but a_n does not converge at all- nonetheless to 1.