A lower bound on $\text{depth }\mathbf{A}\mathbf{/}{\mathbf{I}}^{\mathbf{n}}$

Let $(A,\mathfrak{m})$ be a Noetherian local ring and let $I\subseteq \mathfrak{m}$ be an ideal in $A$. Let $\mathrm{\ell }(I)$ be the analytic spread of $I$. Let $G_I(A)={⨁}_{n\ge 0}{I}^n\mathrm{/}{I}^{n+1}$ be the associated graded ring of $I$ and let its depth be $t$. We prove $\text{depth }A\mathrm{/}{I}^n\ge t-\mathrm{\ell }(I)$ for all $n\ge 1$.