Properties of k-th Roots: Inverse to Powers, Absolute Values, Products, Monotonicity, Subadditivity, Differences and Limits

In the setting of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness, let roots and square roots be as in The k-th Root and the Square Root of a Nonnegative Real Number §root and The k-th Root and the Square Root of a Nonnegative Real Number §square-root, and sequences and convergence as in Sequences §sequence and Convergent Sequences of Real Numbers §converges. Let $k\in\mathbb{N}$ and $x,y\in\mathbb{R}$ with $x\ge0$ and $y\ge0$.

¶ $(\sqrt[k]{x})^{k}=x$ and $\sqrt[k]{x^{k}}=x$; moreover $\sqrt[k]{0}=0$ and $\sqrt[k]{1}=1$.

¶ $\sqrt{z^{2}}=|z|$ for every $z\in\mathbb{R}$.

¶ $\sqrt[k]{xy}=\sqrt[k]{x}\,\sqrt[k]{y}$.

¶ $x\le y$ if and only if $\sqrt[k]{x}\le\sqrt[k]{y}$, and $x<y$ if and only if $\sqrt[k]{x}<\sqrt[k]{y}$.

¶ $\sqrt[k]{x+y}\le\sqrt[k]{x}+\sqrt[k]{y}$.

¶ $|\sqrt[k]{x}-\sqrt[k]{y}|\le\sqrt[k]{|x-y|}$.

¶ If $(a_{n})$ is a sequence in $\mathbb{R}$ with $a_{n}\ge0$ for every $n\in\mathbb{N}$ and $a_{n}\to x$, then $\sqrt[k]{a_{n}}\to\sqrt[k]{x}$.