Limit Superior and Inferior

Real Analysis · Axiom Academy

LESSON Limit Superior and Inferior Watch the tail-supremum fall and the tail-infimum rise — and are where they meet. 1. A Sequence That Won't Settle Take . The even terms sit a little above 1 and slide down toward it; the odd terms sit a little below -1 and climb up toward it. The sequence never converges — but watch where it eventually lives . Pick a cutoff n and look only at the tail . Its supremum is b_n and its infimum is c_n . As the cutoff moves right , the tail shrinks, so b_n can only decrease and c_n can only increase — they squeeze the tail's range inward. The limits they reach are the limsup and liminf. tail-sup b_n decreases to the limsup tail-inf c_n increases to the liminf A smaller set can't have a larger supremum, so — a decreasing sequence. Bounded below, it must converge. A smaller set can't have a smaller infimum, so — an increasing sequence. Bounded above, it must converge. 3. Converges the Band Collapses Always , and the gap between them measures how much the sequence oscillates forever. Shrink that gap to zero and the sequence is trapped: it has nowhere left to go but a single point. Here is the convergent sequence — watch its tail-sup and tail-inf both squeeze to the same value. A genuine gap remains. The sequence keeps oscillating and does not converge (our step-1 sequence: gap = 1-(-1) = 2 ). The band collapses to L . The tail is pinched to a point, so (here gives L = 0 ).

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