Basic question about commutation in Dirac notation

In a nutshell my question is: "In the bracket notation, we can operate in the bra and in the ket with the same operator like $$langle a|(A|brangle)=(langle a|A)|brangle~?$$ So this operations are correct?
$$A|b_irangle=sqrt{b_i}|b_i-1rangleleftrightarrow langle b_i| A^dagger=(sqrt{b_i})^*langle b_i-1|;$$

$$A^dagger|b_irangle=sqrt{b_i+1}|b_i+1rangleleftrightarrow langle b_i|A=(sqrt{b_i+1})^*langle b_i+1|; $$
then applying the first equation above in the Ket $$ langle b_i|A|b_i+1rangle=langle b_1|big(A|b_i+1ranglebig)=langle b_1|big(sqrt{b_i+1}|b_irangle big)=sqrt{b_i+1} $$
Now applying the second relation in the Bra
$$ langle b_i|A|b_i+1rangle=big(langle b_i|Abig)|b_i+1rangle= (sqrt{b_i+1})^*langle b_i+1|b_i+1rangle=(sqrt{b_i+1})^* $$
For that to be true it’s required that the values $in mathcal{R}$.