The mean value theorem gives $|\sqrt{x} - \sqrt{y}| \le \tfrac{1}{\iota(2)} |x - y|$ for $x, y \ge 1$, so the square root is Lipschitz with constant $1/2$ on $[1,\infty)$

Given: The set $I := [1,\infty)$, order-convex with at least two elements (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length), and the function $s : I \to \mathbb{R}$, $s(b) := b^{1/2}$.

Derivative of the square root (For a natural $n \ge 1$, the derivative of $x \mapsto x^{1/n}$ on $(0,\infty)$ is $\frac{1}{\iota(n)}x^{1/n - 1}$, obtained from the inverse rule applied to $x \mapsto x^{n}$; in particular $(\sqrt{x})' = 1/(\iota(2)\sqrt{x})$, at $n = 2$): the map $u \mapsto u^{1/2}$ on $(0,\infty)$ is differentiable at every $b > 0$ with derivative $\frac{1}{\iota(2)}b^{-1/2}$.

Restriction of the domain (The derivative $f'(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c}$ of $f : A \to \mathbb{R}$ at a point $c \in A$ that is a limit point of $A$, and differentiability on a set): $I \subseteq (0,\infty)$, every point of $I$ is a limit point of $I$ (Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$, Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length), and a function differentiable at such a point remains differentiable there after restriction, with the same derivative.

A function differentiable at a point is continuous there (A function differentiable at $c$ is continuous at $c$).

Rational powers (Laws of rational exponents, Monotonicity of $r \mapsto a^{r}$ and of $a \mapsto a^{r}$, Existence and uniqueness of $n$-th roots: a unique $a^{1/n} \ge 0$ with $(a^{1/n})^n = a$): $a^{r} > 0$ for $a > 0$; $a^{-r} = 1/a^{r}$; $1^{1/2} = 1$, since $1 \ge 0$ and $1^{2} = 1$ and the nonnegative square root is unique; and for rational $t > 0$, $a > 1$ implies $a^{t} > 1$ (claim 3 of the monotonicity lemma).

Order arithmetic (Inverses of positives are positive, and reciprocation reverses order, Sign rules for products and monotonicity of multiplication, Canonical naturals are positive and strictly increasing, Multiplying inequalities of positives): $\iota(2) > 0$, so $1/\iota(2) > 0$ and $\iota(2) \ne 0$; $0 < a < b$ gives $0 < 1/b < 1/a$ (Inverses of positives are positive, and reciprocation reverses order); a product of two positive reals is positive (Sign rules for products and monotonicity of multiplication); and the NONSTRICT multiplication of inequalities between nonnegatives, $0 \le x \le y$ and $0 \le u \le v$ imply $xu \le yv$, is Multiplying inequalities of positives and is not stated by Sign rules for products and monotonicity of multiplication, whose multiplicative claims are strict. Also $|u| = u$ for $u \ge 0$ (Basic properties of the absolute value).

Interiority (Interior, closure, boundary and exterior of a subset of $\mathbb{R}$, The $\varepsilon$-neighbourhood and the punctured $\varepsilon$-neighbourhood of a point of $\mathbb{R}$): $p$ is interior to $S$ exactly when $N_{\varepsilon}(p) \subseteq S$ for some real $\varepsilon > 0$.

Bounded derivative gives Lipschitz (If $f$ is continuous on an interval $I$ and $|f'| \le M$ at every interior point, then $|f(x) - f(y)| \le M|x-y|$ for all $x,y \in I$, so $f$ is Lipschitz with constant $M$ and uniformly continuous on $I$): for $J$ order-convex, $h : J \to \mathbb{R}$ continuous on $J$ and differentiable at every interior point of $J$, and a real $M \ge 0$ with $|h'| \le M$ at every interior point, one has $|h(x)-h(y)| \le M|x-y|$ for all $x, y \in J$; such an $h$ is Lipschitz with constant $M$ and uniformly continuous on $J$ (Lipschitz map, $\alpha$-Hölder map for rational $0 < \alpha \le 1$, and contraction, Dictionary: for $A \subseteq \mathbb{R}$ with the metric $d(x,y) = |x-y|$, continuity and uniform continuity of $f : A \to \mathbb{R}$ agree with the metric-space notions, the Lipschitz and Hölder conditions are the metric ones instantiated, and a subset of $\mathbb{R}$ is compact in the open-cover sense of $\mathbb{R}$ exactly when it is a compact metric subspace, Uniform continuity of $f : A \to \mathbb{R}$: one $\delta$ serving every pair of points of $A$).