Let $E$ is ellipsoid in $\mathbb{R}^n$.

Let $$E'$$ is ellipsoid of dimension $$n-1$$ that gain as intersection of $$E$$ and some hyperplane. Let $$a_1\leq\cdots\leq a_n$$ are halfaxis of $$E$$ and $$b_1\leq\cdots\leq b_{n-1}$$ are halfaxis of $$E'$$. Prove that for any $$i$$ from $$1,\cdots,n-1$$ is truth that $$a_i\leq b_i\leq a_{i+1}.$$

It is obviously for $$n=2$$. But I can't do induction.