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## Main Question or Discussion Point

Let

*(x*be a bounded sequence that diverges. Show that there is a pair of convergent subsequences_{n})*(x*and_{nk})*(x*, so that_{mk})[tex]

lim_{k\rightarrow\infty}

[/tex] [tex]\left|x_{nk} - x_{mk}\right| > 0[/tex]

lim_{k\rightarrow\infty}

[/tex] [tex]\left|x_{nk} - x_{mk}\right| > 0[/tex]