Finding Convergent Subsequences

Real Analysis · Axiom Academy

EXAMPLE Finding Convergent Subsequences of a_n = (-1)^n Bolzano–Weierstrass in action: pulling two convergent subsequences out of a sequence that has no limit The sequence a_n = (-1)^n is bounded but does not converge. Exhibit two explicit convergent subsequences and find each of their limits. The two subsequences at a glance Every term sits at either +1 or -1 . The even-index terms (green) form a constant subsequence at +1 ; the odd-index terms (red) form a constant subsequence at -1 . You extracted two convergent subsequences from a sequence that has no limit of its own. Divergence is not the end of the story: a_n=(-1)^n oscillates and has no limit, but it is bounded. Index patterns build subsequences: the even indices give (a_ 2k ) and the odd indices give (a_ 2k-1 ) . Two different limits: and , so the subsequential limits are . Bolzano–Weierstrass: every bounded sequence has a convergent subsequence — here the bound forces at least one to exist, and we found two. This trick — split by an index pattern, then read off the constant value — is the workhorse behind sequential compactness and the Bolzano–Weierstrass theorem.

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