Does the Maximum of Two Convergent Series Also Converge?

[porroadventum]

Homework Statement

Let Ʃ from n=1 to ∞ a_n and Ʃ from n=1 to ∞ b_n be convergent series, with a_n\geq0 and b_n\geq0 for all n\inN. Show that the series Ʃ from n=1 to∞ max(a_n,b_n) converges.

Homework Equations

I'm guessing it's got something to do with the cauchy criterrion for convergence of series but I'm not sure where to begin? Any hints would much appreciated

 

[lurflurf]

I would write

2 max(a,b)=a+b+|a-b|

or

max(a,b)<=a+b

Either of which obviously converge.