limsup inequality

I need to show that limsup(an + bn) ≤ lim sup(an) + lim sup(bn) but my question is not actually how this is proved.

I have seen that when proving the identity lim sup an=− lim inf −an
many sources use the identity sup(A)= - inf(-A) where A is a bounded set
Here is one;
"Indeed, using sup(A) = − inf({−a : a ∈ A}), which follows as if x is an upperbound for A, then −x is an lowerbound for {−a : a ∈ A}, and vice versa, we have lim sup an = limn→∞ sup{am : m > n} = limn→∞ − inf{−am : m > n} = − lim inf −an"

Now my question is that why can't we use the identity sup(A+B) = sup(A)+sup(B) to conclude that limsup(an + bn) "=" lim sup(an) + lim sup(bn) but instead the result is limsup(an + bn) ≤ lim sup(an) + lim sup(bn)... isn't sup{am+bm : m>n} taking the supremum of a set as well as sup{am : m > n} and sup{bm : m > n} just like the proof for lim sup an=− lim inf −an and use sup(A+B)=sup(A)+sup(B)

I need to show that limsup(an + bn) ≤ lim sup(an) + lim sup(bn) but my question is not actually how this is proved.

I have seen that when proving the identity lim sup an=− lim inf −an
many sources use the identity sup(A)= - inf(-A) where A is a bounded set
Here is one;
"Indeed, using sup(A) = − inf({−a : a ∈ A}), which follows as if x is an upperbound for A, then −x is an lowerbound for {−a : a ∈ A}, and vice versa, we have lim sup an = limn→∞ sup{am : m > n} = limn→∞ − inf{−am : m > n} = − lim inf −an"

Now my question is that why can't we use the identity sup(A+B) = sup(A)+sup(B) to conclude that limsup(an + bn) "=" lim sup(an) + lim sup(bn) but instead the result is limsup(an + bn) ≤ lim sup(an) + lim sup(bn)... isn't sup{am+bm : m>n} taking the supremum of a set as well as sup{am : m > n} and sup{bm : m > n} just like the proof for lim sup an=− lim inf −an and use sup(A+B)=sup(A)+sup(B)

For a counterexample to equality

let $(a_n)$ be the sequence

0,1,0,1,0,1,0,1,...

and let $(b_n)$ be the sequence

1,0,1,0,1,0,1,0,...

Edit: I suspect you were unconsciously assuming the sequences converged.

No it is not that I was assuming that the sequences converge... But i can now see what I was thinking wrong, I was treating the sequences as sets, which can be "sometimes" useful in other concepts but not for this case. so we have sup(A+B)=supA+supB if A+B={a+b∣a∈A,b∈B} but with limsup(an + bn) an+bn is not equal to {a(i)+b(j)∣a(i)∈an,b(j)∈bn}. so with the sequences the addition refers to term by term addition and this is where I was careless.

No it is not that I was assuming that the sequences converge... But i can now see what I was thinking wrong, I was treating the sequences as sets, which can be "sometimes" useful in other concepts but not for this case. so we have sup(A+B)=supA+supB if A+B={a+b∣a∈A,b∈B} but with limsup(an + bn) an+bn is not equal to {a(i)+b(j)∣a(i)∈an,b(j)∈bn}. so with the sequences the addition refers to term by term addition and this is where I was careless.