Is it true that $\sup\{|a+b|\} \leq \sup\{|a| + |b|\}$?

Question

Is it true that $\sup\{|a+b|\} \leq \sup\{|a| + |b|\}$? I am trying to argue that the statement in the last sentence is true and all I can think about to argue for it is that we know that $|a+b| \leq |a| + |b|$ and so the $\sup$ of the left side will always be less than or equal than the right hand side. Can anyone tell me if this reasoning is enough to justify the inequality or if I need to think about something else?

Answer 1

Show that the right side is an upper bound on $\{|a+b|\}$. Since the $sup$ is the least upper bound you will have proven the inequality.