questions on bounds??

[Proff]

For any a < b, and for any two functions f and g defined on [a, b] which are bounded above on [a, b], prove that, sup{( f + g)(x) ∶ x∈ [a,b ]} ≤ sup{ f(x) ∶ x ∈ [a, b]} + sup{ g(x) ∶ x∈[a,b]}. Giving an example, show that the strict inequality of the above claim can hold.

How to solve this? I find bounds hard

 

[mathmari]

\(\displaystyle f(x)\leq sup(f(x)), \forall x\epsilon [a,b] \)
\(\displaystyle g(x)\leq sup(g(x)), \forall x\epsilon [a,b] \)
\(\displaystyle f(x)+g(x)\leq sup(f(x))+sup(g(x)), \forall x\epsilon [a,b] \)
\(\displaystyle sup(f(x))+sup(g(x)) \) is an upper bound of \(\displaystyle (f+g)(x) \)
The supremum is less than or equal to any upper bound
so \(\displaystyle sup(f+g)\leq sup(f)+sup(g) \)

 

[daon2]

As a hint for your example, look for two functions which when added cancel but that aren't constant.