I have the following problem. Hope someone can help me. Consider for $N \in \mathbb{N}$ the space $\mathbb{C}^{N}$ of all vectors of length $N$. Now define for $p > 0$ the following norm
$$ ||x||_{p,\infty} := \inf\bigl\{M \geq 0 : \text{card}\left(\{ j \in [N] : |x_{j}| \geq t\} \right) \leq \frac{M^{p}}{t^{p}} \text{for all } t > 0 \bigr\}. $$
Now $||\cdot||_{p,\infty}$ defines a quasinorm, i.e. it satisfies the norm axioms, except that the triangle inequality is replaced by
$$ ||x + y||_{p,\infty} \leq K\left(||x||_{p,\infty} + ||y||_{p,\infty}\right) $$ for some $K > 0$.
I want to show the second norm axiom, i.e. $$ ||\lambda x||_{p,\infty} = |\lambda| \cdot ||x||_{p,\infty} $$ for all $\lambda \in \mathbb{C}$. Maybe the question is trivial, but how can I formally prove that the above quasinorm satisfies the second axiom?
For nonzero $\lambda$ we have \begin{align} ||\lambda x ||_{p, \infty} &= \inf \{ M \geq 0 : \operatorname*{card}(\{ j \in [N] : |\lambda x_j| \geq t\}) \leq \frac{M^p}{t^p}\,,\forall t > 0\} \\ &= \inf \{ |\lambda| \frac{M}{|\lambda |} \geq 0 : \operatorname*{card}(\{ j \in [N] : |x_j| \geq \frac{t}{|\lambda|} \}) \leq \left(\frac{|\lambda |}{t} \right)^p \left(\frac{M}{|\lambda|} \right)^p \,, \forall t > 0\} \\ &= |\lambda| \inf \{ M' \geq 0 : \operatorname*{card}(\{ j \in [N] : |x_j| \geq t' \}) \leq \frac{M'^p}{t'^p} \,, \forall t' > 0\} \\ &= |\lambda|\, || x ||_{p,\infty}. \end{align}