The following prop. is quoted from Folland's Real Analysis. I understand every part of the proof except for $$|x^\beta|\leq(1+|x|)^N$$ and the very last inequality involving $\delta^{-1}$ in the line after "... and hence". Could you give me any idea of how to show the inequalities?
$$|x^\beta| = |x_1^{\beta_1}\cdots x_n^{\beta_n}| \le |x|^{\beta_1}\cdots |x|^{\beta_n} = |x|^{\beta_1 + \cdots \beta_n} = |x|^{|\beta|}.$$ As for your last question, it looks like that last term should be $$2^N(1+\delta^{-1})\sum_{|\beta|\le N}|x^\beta|,$$ the reason being that $1,|x_1|^N,\dots , |x_n|^N$ are all included in the sum, but we don't know which is bigger, $1$ or $\delta^{-1}.$