Forward Euler has uniform local order one and a one-step Lipschitz bound

Given: The stated open ODE domain with open convex state fibers, continuous first partial derivatives and bounds, the solution through $(0,y_0)$ on an open interval containing $[0,T]$, the compact set $K$ containing its graph and the Euler time-state pairs under consideration, and the definition of uniform local order.

One-step ODE methods, local defect, uniform order, and step stability: ${\delta }_h(t;y)={\mathrm{\Psi }}_h(t,y(t))-y(t+h)$; uniform local order one means a uniform $O(h^2)$ defect; one-step Lipschitz stability is the displayed $1+Lh$ bound.

First-order systems, initial value problems, and solutions on intervals: an IVP has open time-state domain, and a solution is differentiable on an interval, has graph in the domain, and satisfies $y^{\mathrm{\prime }}=f(t,y)$ and its initial value.

If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative: continuous first partial derivatives give a total derivative whose matrix is the Jacobian of those partials.

Total differentiability gives a local $O(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ increment bound and therefore continuity: total differentiability implies continuity at each point.

A convex subset of ${\mathbb{R}}^m$ contains every line segment between two of its points: a convex set contains the segment between each pair of its points; each open convex state fiber has this property.

On a convex open set, a uniform bound $\mathrm{\parallel }Df(z)v{\mathrm{\parallel }}_2\le M\mathrm{\parallel }v{\mathrm{\parallel }}_2$ implies $\mathrm{\parallel }f(y)-f(x){\mathrm{\parallel }}_2\le M\mathrm{\parallel }y-x{\mathrm{\parallel }}_2$: on an open convex Euclidean set, a uniform derivative bound $M$ implies $\mathrm{\parallel }g(u)-g(v){\mathrm{\parallel }}_2\le M\mathrm{\parallel }u-v{\mathrm{\parallel }}_2$.

Every Picard–Lindelöf initial value problem has one maximal solution on an open interval: a continuous vector field locally Lipschitz in the state variable has a unique maximal IVP solution. The given derivative bound and [F3]–[F6] provide that local Lipschitz condition on each state fiber.

A vector field along an ODE solution has an explicit derivative bound: along an exact solution, $g_i(s)=f_i(s,y(s))$ has derivative bounded by $B_i=M_{t,i}+{\sum }_{k=1}^d{M}_{ik}{F}_k$.

A first-order initial value problem is equivalent to its Volterra integral equation: for any initial time $t$ on the solution interval, the solution satisfies $y(t+h)=y(t)+{\int }_t^{t+h}f(s,y(s))ds$.

The mean value theorem, as the case $g(x)=x$ of Cauchy's: for $f$ continuous on $[a,b]$ with $a<b$ and differentiable on $(a,b)$ there is $c\in (a,b)$ with $f(b)-f(a)=f^{\mathrm{\prime }}(c)(b-a)$: if a real function is continuous on a closed interval and differentiable in its interior, its endpoint difference is its derivative at an interior point times the interval length.

The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$: for $x\in {\mathbb{R}}^d$, $\mathrm{\parallel }x{\mathrm{\parallel }}_2^2={\sum }_{i=1}^d{x}_i^2$.

Cauchy-Schwarz $\mid ⟨x,y⟩\mid \le \parallel x{\parallel }_2\parallel y{\parallel }_2$ with its equality case, the triangle inequality for $\parallel \cdot {\parallel }_2$, the parallelogram law and polarisation: for $x,z\in {\mathbb{R}}^d$, the Euclidean norm satisfies $\mathrm{\parallel }x+z{\mathrm{\parallel }}_2\le \mathrm{\parallel }x{\mathrm{\parallel }}_2+\mathrm{\parallel }z{\mathrm{\parallel }}_2$.