What is $sup{ (-1)^n + 1/n} $? My doubt is that $\sup (A+B) = \sup A + \sup B$.
So, if we consider  $A =(-1)^n$ and $B=\frac{1}{n},$
then $\sup A = 1 $  and   $\sup B = 1$.
So will $\sup (A+B) = \sup{ (-1)^n + 1/n}  = \sup (-1)^n   +  \sup \frac{1}{n} = 1+1 =2 ?$
Is this correct or have I misunderstood somewhere please explain?
Also could you please provide the correct method to solve the above problem?

The problem here is that in the statement
$$
\sup (A + B) = \sup A + \sup B
$$
the definition of "$+$" is given by 
$$
A+B=\{a+b\mid a\in A, b\in B\},
$$
and $A$ and $B$ are sets (of real numbers)
. 
You, in trying to apply it, appear to be adding two sequences termwise, i.e., you're defining a new sequence 
$$
(A+B)_n = A_n + B_n
$$
With this definition of addition (and the implied notion that "sup" of a sequence is the sup of the set of items in the sequence) the "theorem" is no longer a theorem, as your counterexample shows. 
