Help needed to understand the proof of Projective Nullstellensatz

I am self studying Algebraic Geometry from the Clader and Ross Algebraic geometry textbook: Beginnings in Algebraic Geometry.

This proof is given on page 275 of the textbook and I am quite confused about it. Please help me.

Theorem 9.53 (Projective Nullstellensatz) Assume that $K$ is algebraically closed. Let $I \subseteq K[x_0, \cdots , x_n]$ be a homogeneous ideal.

1. $V_{\mathbb{P}}(I) = \varnothing$ if and only if $ \langle x_0, \cdots , x_n\rangle\subseteq \sqrt{I}$
2. If $\sqrt{I}$ is relevant then $I_{\mathbb{P}}(V_{\mathbb{P}}(I)) =\sqrt{I}$

In proof of the $1$st assertion I am not able to understand that if $V_{\mathbb{P}}(I)=\varnothing$ and $I= K[x_0,\cdots,x_n]$ then how does it follows that $\sqrt{I}= K[x_0,\cdots,x_n] \supseteq\langle x_0,\cdots,x_n\rangle$.

Now suppose that $\sqrt{I}$ is relevant. Let $V_{\mathbb{P}}(I)= \varnothing$. Then I understand how $\sqrt{I} \supseteq \langle x_0,\cdots,x_n\rangle$ but why the only relevant ideal containing the (maximal) irrelevant ideal should be the full ring $K[x_0,\cdots,x_n].$

Rest of the proof is clear to me.

Kindly help me with these 2 questions!