Equalities involving power of natural numbers

Question

If I have $N\in\mathbb{N}$ and $\alpha>0$ is the following true? $$(2N)^\alpha-(2N+2)^{\alpha}<0$$ and $$-(2N+1)^\alpha+(2N+3)^{\alpha}>0$$ I think yes since $2N<2N+2$ and $2N+3>2N+1$.

Answer 1

Yes those are true this because for $x\ge 0$ , $x^\alpha$ for any constant $\alpha>0$ is a monotonically increasing function of $x$ which means that $a>b\implies a^{\alpha}>b^{\alpha}$ for all $a,b\ge 0$.