Could you give an example of the usage? Usually it is used when you have two convergences, say for each epsilon we have .

Now to conclude that x_n+y_n converges to x+y you have

the first one on the RHS is less than epsilon if n>N_1, and the second is less than epsilon if n>N_2. To get both of them to be less than epsilon at the same time, you must take the maximum, otherwise you cannot know if both are less than epsilon.

Hope this was what you were looking for.