In Hartshorne's "Algebraic Geometry", the following statement is a weaker form of Theorem 8.21A (e), which he quotes from Matsumuura's book on commutative algebra:

Proposition.Let $R$ be a regular local ring and $I=(x_1,\ldots,x_r)\subset R$ an ideal generated by a regular sequence. Let $A:=R/I$. Then, $$\begin{eqnarray*} \phi: A[T_1,\ldots,T_r] &\overset{_\sim}{\longrightarrow} & \mathrm{gr}_I(R) = \bigoplus\nolimits_{d\ge 0} {I^d}/{I^{d+1}} \\ T_i & \longmapsto & x_i \end{eqnarray*}$$ is an isomorphism of graded $A$-algebras.

In Hartshorne, the condition of being regular and local is strengthened to Cohen-Macaulay. However, I only need the above. I tried to look up the proof for the general statement in Matsumuura's book, but it seems rather involved (and honestly, a bit convoluted). I would like to use results about regular local rings and in turn, avoid introducing terminology like Hilbert-Samuel polynomials.

So, I guess I am asking for an "easy" proof of the above proposition. It seems rather easy for $r=1$, but I am somehow stuck trying to prove it by induction.