Look at |f(x)+g(x)- (a+b)|= |(f(x)-a)+(g(x)-b)| and use the fact that . In order to make , we have to make sure that |f(x)-a| and |g(x)-b| add up to something less than . One way to do that is to make sure that |f(x)-a| and |g(x)-b| each are less than . Since , there exist some such that if [tex]|x- x_0|< \delta_1[\math], . Since , there exist some such that if , .
To be sure that both of those happen, choose to be less than or equal to the smaller of and . That way, if , both and are true.