Consider a separable Hilbert space $\mathcal{H}$, we can first define $T(\mathcal{H})$ the trace class of $\mathcal{H}$, then $D(\mathcal{H})$ to denote the interesection of positive operator and trace class with trace no more than 1.

For $A,B\in D(\mathcal{H})$, we say $A\leq B$ if $A-B$ is positive, or (A,B) satisfies the Lowner order.

Now suppose we have increasing sequence $$ A_0,A_1,\cdots,A_n,\cdots $$ with $(A_i,A_{i+1})$ satisfies the Lowner order, and $A_i\in D(\mathcal{H})$.

Does this sequence has least upper bound? In other words, does this class has limit?