proving a inequality about sup 
Possible Duplicate:
How can I prove $\sup(A+B)=\sup A+\sup B$ if $A+B=\{a+b\mid a\in A, b\in B\}$ 

I want to prove that  $\sup\{a+b\}\le\sup{a}+\sup{b}$ and my approach is that I claim $\sup a+ \sup b= \sup\{\sup a + \sup b\}$ and since $\sup a +\sup b \ge a+b$ the inequality is proved. Is my approach correct?

It is better to think about what the inequality says than just doing formal manipulations.  On the left-hand side you have less freedom than on the rigt-hand side, this is easier to see if you write the argument of $a$ and $b$ explicitely, as such:
$$ \sup_x\{a(x)+b(x)\} \le \sup_x a(x) + \sup_x b(x)
$$
On the LHS, you must vary $x$ simultaneously in $a$ and in $b$, on the RHS you can vary $x$ separately in $a$ and in $b$. In that way, every value you can get on the LHS can be matched on thje RHS, but not in the opposite direction. That makes the inequality obvious.
