# Math Help - Real analysis - limit proofs

1. ## Real analysis - limit proofs

Let a(n) and b(n) be convergent sequences of real numbers such that the limit of a(n) as n approaches infinity is A and the limit of b(n) as n approaches infinity is B for some real numbers A and B. Show (using the epsilon-N definition of a limit) that

(a) the limit of a(n) + b(n) as n approaches infinity = A + B

(b) the limit of a(n)b(n) as n approaches infinity = AB

2. The following are hints.

Originally Posted by alexmin
(a) the limit of a(n) + b(n) as n approaches infinity = A + B
$|a_n + b_n - (a+b)| = |(a_n-a)+(b_n-b)| \leq |a_n-a|+|b_n-b|$

(b) the limit of a(n)b(n) as n approaches infinity = AB
Since $a_n \to a$ the sequence is bounded, so $|a_n|\leq M$ for $M>0$.
This means,
$|a_nb_n - ab| = |a_nb_n - a_nb + a_nb - ab| \leq |a_n||b_n - b|+ |b||a_n - a| \leq M|b_n-b|+b|a_n-a|$.