In what sense, if any, can we construct an unbounded operator as a limit of an unbounded sequence of bounded operators?
A matrix representation of an algebra of raising and lowering operators, for example, is often displayed, in effect, as the limit of an unbounded sequence of bounded matrix operators, but in what sense is the limit close to the raising and lowering operators?
Edit: By "raising and lowering operators", I mean an algebra generated by $a^\dagger$ and $a$ that satisfies the commutation relation $[a,a^\dagger]=1$. If someone answers the specific question of the second paragraph, I hope I'll be able to reconstruct some kind of answer to the more abstract question of the first paragraph from that.
For operators that have well defined resolvents (eg. self-adjoint, M-sectorial, operators with discrete spectra) you can work with various types of convergence of the resolvents (norm, strong, weak). The raising and lowering operators are not nice. You can work with $a + a^\dagger$ or $i(a - a^\dagger)$.