Page 154 - 'Blast_Into_Math

Blast into Math! Analatic nummer theora: ants, ghosts and giants

Using only the definition of limit, we can prove some important facts about limits. This proposition will
be useful to shed light on many proofs and problems.

Proposition 7.1.3 (Limit Addition and Multiplication Proposition (LAMP))

∞ be a sequence which converges to a limit X . For any number s, the sequence
1. Let {x n } n=1
∞
{s n } n=1 ,

with s n = x n + s converges to X + s .

2. The sequence
∞ ,
{t n } n=1 t n = sx n

converges to sX .

∞ ∞ with
3. If another sequence {y n } n=1 converges to a limit Y , then the sequence {z n } n=1
z n = x n + y n converges to X + Y .

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