Let $R$ a ring and $M$ be an $R-$module freely generated by a set $X$, and let $Y$ be a subset of $X$. Show that $Y$ freely generates $RY$

Question

I have tried the following way: I want to show that $RY=L_R(Y)$, then be $rz\in RY$ where $r\in R$ and $z\in Y$, as $Y\subseteq X\subseteq M$ and $M=L_R(X)$ then $z= \sum_{i=1}^{n}r_ib_i$ where $X=\left \{ b_1,b_2,...,b_n\right \}$ so $rz=\sum_{i=1}^{n}(rr_i)b_i$. But I need is that every $b_i\in Y$ and I do not know how to do this.