Algebra of Limits

Real Analysis · Axiom Academy

If and , then — limits pass straight through sums, products, and quotients. Take two convergent sequences and add them term by term. The new sequence a_n + b_n converges, and its limit is exactly the sum of the two limits — the combined track settles to A + B at the same moment the parts settle to A and B . The same idea covers differences and scalar multiples Multiplying term by term works too: . Picture the product as the area of a rectangle whose sides are a_n and b_n . As the sides settle toward A and B , the area settles toward . a_n b_n - AB = a_n(b_n - B) + B(a_n - A) — each piece is small. Dividing term by term gives — provided . That hypothesis is the whole story: when , the terms b_n stay bounded away from the forbidden zero line, so the division never destabilizes. Why matters: because , there is an N past which . The denominators are bounded away from zero, so 1/b_n is well-behaved and stays close to 1/B . b_n settles away from 0 ; the quotient a_n/b_n converges to A/B . Dividing by terms near 0 can blow up or oscillate — the limit need not exist. The limit operator distributes over algebra: once two sequences converge, their sum, difference, scalar multiple, and product converge to the combined values — and so does the quotient, as long as the bottom limit isn't zero. Scroll up to revisit any step.

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