What does it mean in convergence/divergence proofs when you see "max(N,N1)?"
Is it that we are not sure where the N will be on the real number line so we will just say the maximum N, N1 to include every possible N, N1 on the real line?
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.