Different Besov-Norm Definitions

first some notation: $\langle x\rangle=\sqrt{1+x^2}$, $P_{j}$ is the Littlewood Paley Projector and $P_{\leq0}$ corresponds to the small frequencies.

I have a the following definition of the Besov norm:

$$||u||_{B_{p,q}^s}= ||P_{\leq0}(f)||+(\sum^\infty_{j=1}(2^{js}||P_j(f)||_{p})^{q})^{1/q}$$ and $$||u||_{B_{p,\infty}^s}= ||P_{\leq0}(f)||+\sup_{j\geq1}2^{js}||P_j(f)||_{p}$$ I was told they were equivalent to the following but I'm not sure (especially about whether I want to allow negative $j$ and whether I have to use $\langle\rangle$ or not) and I'm failing to prove the equivalence of the Norms: $$||u||_{B_{p,q}^s}= (\sum^\infty_{j=-\infty}(\langle2^{j}\rangle^s||P_j(f)||_{p})^{q})^{1/q}$$ and $$||u||_{B_{p,\infty}^s}= \sup_{j\in\mathbb{Z}}\langle2^j\rangle^{s}||P_j(f)||_{p}$$

My question is: Did I get the second definitons right and how can I prove equivalence?

Furthermore it is obvious (since all norms on $\mathbb{R}^n$ are equivalent) that: $$||P_{\leq0}(f)||+(\sum^\infty_{j=1}(2^{js}||P_j(f)||_{p})^{q})^{1/q} \sim (\sum^\infty_{j=0}(2^{js}||P_j(f)||_{p})^{q})^{1/q}$$
IF by definition $P_{\leq0} = P_{0}$. This is also explained in the referenced paper.