Let $(R, \mathcal{m},K)$ be a local ring and $J_0\supseteq J_1 \supseteq J_2\supseteq \cdots$ are all reductions of $I$. Then $\bigcap_{n\ge 0}J_n$ is also a reduction of $I$.
I have done.
I take the family $\{\frac{J_i+\mathcal{m}I}{\mathcal{m}I}\}$. Each element has finite dimension as vector space over $K$. Then there exists $N$ nonnegative integer such that $\frac{J_N+\mathcal{m}I}{\mathcal{m}I}=\frac{J_i+\mathcal{m}I}{\mathcal{m}I}$ for $i \geq N$, and so $J_N+\mathcal{m}I=J_i+\mathcal{m}I$ for $i \geq N$.