I am trying to prove for $K,L \in H^*$ that $K+L \in H^*$
In the solution, we first take a sequence $(u)j)_j \subset H$ such that $u_j \to u $ in $H$.
Then $$\lim_{j \to \infty} (K+L)(u_n)$$ $$= \lim_{j \to \infty} (K u_n + L u_n)$$ $$=\lim _{j \to \infty} K u_n + \lim _{j \to \infty} Lu_n$$ $$= Ku + Lu$$ $$= (K+l)(u)$$
My question is, why do we need to involve a sequence $(u_j)_j$?
Can't the proof still work without using the sequence and limits?