# Math Help - whats the rule about..

1. ## whats the rule about..

whats the rule about the sum of to convergent sequences??

2. Originally Posted by transgalactic
whats the rule about the sum of to convergent sequences??
Please, can't you be more careful when you write something ???

it's not "to", it's "two"... without context, it would be hard to guess

Let $a_n \to a$ : this means that $\forall \epsilon>0, \exists N, \forall n>N, |a_n-a|<\epsilon/2$
similarly, $b_n \to b$ : $\forall \epsilon>0, \exists N', \forall n>N', |a_n-a|<\epsilon/2$

now what about $a_n$ and $b_n$ ? do they converge to $a+b$ ?

$|a_n+b_n-(a+b)|=|(a_n-a)+(b_n-b)| \leq |a_n-a|+|b_n-b|$, by the triangle inequality.
And for $n>\max(N,N')$, $|a_n-a|+|b_n-b|<\epsilon/2+\epsilon/2=\epsilon$

this is the definition for $a_n+b_n \to a+b$