Pet Peeve (notation for limits assumes uniqueness...)

When I teach limits in advanced calculus, the first step is teaching the definition of convergence of a sequence:

Definition: Given a sequence of real numbers $(x_n)$ and a real number $L$, the sequence $(x_n)$ converges to $L$ if [... we all know what goes here ...]

After a few simple examples, I immediately prove:

Theorem: Given a sequence of real numbers $(x_n)$ and real numbers $L,M$, if the sequence $(x_n)$ converges to $L$ and the sequence $(x_n)$ converges to $M$ then $L=M$.

and then I define

Definition: $\lim_{n \to \infty} x_n$ is equal to the unique number to which $(x_n)$ converges.

where the previous theorem is used to justify uniqueness, and therefore well-definedness.

I see that my textbook does indeed follow this same line of development: Advanced Calculus, by Patrick M. Fitzpatrick.