I need help in finding the mistake in the following reasoning.
I proved that if dimension of Banach space $X$ is infinite, then multiplication of bounded operators is separately continuous but not continuous in strong operator topology. This fact can be found in many sources, e.g. Halmos or Simon & Reed so I know it is true.
It is also claimed (e.g. in Halmos) that multiplication is sequentially continuous. Problem is that I think proof I came up with also works for nets, which would contradict statement above. I present it below. I need help with finding error.
Suppose $A_i \to A$ and $B_i \to B$ strongly. Then both nets are bounded by uniform boundedness principle. In particular $\| A_i \| \leq M$. Furthermore we have:
$AB-A_iB_i=(A-A_i)B+A_i(B-B_i)$
Therefore:
$\|(AB-A_iB_i)x\| \leq\|(A-A_i)Bx\|+\|A_i\|\|(B-B_i)x\|$
Which converges to zero, so $A_i B_i \to AB$.
Nets that converge do not need to be bounded. Even when you use the uniform boundedness principle, you see that the operators applied to individual elements need not be bounded. For an example of this, imagine a net that is indexed by two copies of $\mathbb{N}$ cascaded. (So every element of one copy is greater than the other.) I believe this is typically called the ordinal sum of two copies of $\omega$.