Suppose $1\le p<q\le\infty.$ We will prove that $\ell^{p}\subseteq \ell^{q}$ by proving that $\|a\|_{q}\le\|a\|_{p}$ for any $a\in \ell^{p}.$ Define $b_n := \frac{a_n}{\|a\|_{p}}$ and observe that $\vert b_n \vert \leq 1$ for all $n \in \mathbb N$. Notice the following chain of equivalent statem...

Suppose $J \subseteq \sqrt I$. We want to show $V(I) \subseteq V(J)$. If you have the following result at your disposal: $\sqrt I = \cap \{\mathfrak p$ prime $ : I \subseteq \mathfrak p\}$, then your reasoning works: Let $\mathfrak p \in V(I)$, then $I \subseteq \mathfrak p$. But since $J \subs...