It is true, for some $N$ constant, that $x^{p}\leq Nx^{\alpha p}$ for $\alpha>1,\ p\geq 1$?

Question

I know that $(x+y)^p\leq 2^p(x^p+y^p)$ for all $x,y\geq 0$ and $p\geq 0$.

Let $\alpha>1$ and $p\geq 1$. Exists a constant $C$ such that, for all $x>0,\ x^{p}\leq N x^{\alpha p}$?

If $x\geq 1$, then $x^{p}\leq x^{\alpha p}$ is holds. If $x<1$, Is it true?

I try something like this $x^p\leq (x+1)^p\leq (x+1)^{\alpha p}\leq 2^p(x^{\alpha p}+1)$ but obviously it doesn't work since there is 1 ...

Answer 1

This is a question but I hope it will suffice as an answer: What is $\displaystyle \lim_{x \to 0^+} x^{p(1-\alpha)}$?

Recall $p(1-\alpha) < 0$.