After some more thought, I proved that 1) is true using proof by contradiction. (I always forgot to use the fact that it's the LEAST upper bound). Then taking limits, I got the result because the inequality holds for ALL epsilon>0, so we can get rid of the epsilons.
How can I prove the other direction? (I am trying to modify my proof but I got stuck)
Given ε>0.
a(n)->a => there exists N1 s.t. if n≥N1, then a(n)>a-ε. <-----this is true for sure
b=limsup b(n) => there exists N2 s.t. if n≥N2, then b(n)> ???
Can somebody help me, please?