Suppose $A$ is a $C^*$-algebra without unit, $A^+$ is a unitization of $A$, treat $A$ in the $A^+$, if $\{x_n\}$ in $A$ converge （or monotonous converge） to $1$ in $A^+$, does $\{x_n\}$ must be the approximate identity in $A$?
If yes，let a is a strictly positive element in $A$, $a^{1/n}$ is approximate identity in $A$, we can prove it from above way ($\{1,a\}$ generates a subalgebra isomorphism with $C(K)$ and $a^{1/n}$ converge to $1$ in $C(K)$), but some books prove it with some long technology related the representation, why?
The norm in $A^+$ is the same as in $A$. So if your sequence $\{x_n\}$ converges to $1$, this means that $1\in A$, since $A$ is complete (as it sits as a closed ideal in $A^+$).
To think about this stuff it is often useful to think of $C_0(\mathbb R)$ as your prototype of non-unital C$^*$-algebra.