constructing grammar for 1*0*1(1+0)*

Question

I want to construct a grammar for the following regular expression: $1^*0^*1(1+0)^*$. I did it the following way:

$S \rightarrow AB1C$

$A \rightarrow 1A | \epsilon$

$B \rightarrow 0B | \epsilon$

$C \rightarrow 0C|1C| \epsilon$

Is it correct?