His name is on a single inequality that says how fast a function may stretch. Put a ceiling below 1 on that rate and iteration is forced to settle on one answer. This page builds that from a chapter of numerical-analysis notes and checks the notes’ worked example.
The notes give him a footnote: Rudolf Otto Sigismund Lipschitz, born 14 May 1832 in Königsberg, Prussia (now Kaliningrad, Russia), died 7 October 1903 in Bonn. It says he made important contributions to number theory, the theory of Bessel functions and Fourier series, ordinary and partial differential equations, and analytical mechanics and potential theory. That is everything this chapter will say about his life. A chapter with more sources would say more.
A function $g$ on an interval $[a,b]$ satisfies a Lipschitz condition with constant $L > 0$ if
When $0 < L < 1$ the function is a contraction: the distance between two images is at most $L$ times the distance between the points, so every pair of points is pulled closer together.
Let $g$ be continuous on $[a,b]$ with $g(x) \in [a,b]$ for all $x$, and a contraction there. Then $g$ has a unique fixed point $\xi$ in $[a,b]$, and for every start $x_0 \in [a,b]$ the iteration $x_{k+1} = g(x_k)$ converges to $\xi$.
Uniqueness. If $\xi$ and $\eta$ were both fixed, then $|\xi - \eta| = |g(\xi) - g(\eta)| \le L|\xi - \eta|$, so $(1-L)|\xi - \eta| \le 0$; since $1 - L > 0$, $\eta = \xi$.
Convergence. $|x_k - \xi| = |g(x_{k-1}) - g(\xi)| \le L|x_{k-1} - \xi|$, and by induction $|x_k - \xi| \le L^k |x_0 - \xi|$, which tends to 0 because $L < 1$.
The source gets existence of a fixed point on the interval from Brouwer’s fixed point theorem (its Theorem 1.2). The uniqueness and the rate need only the Lipschitz condition. The source does not say who first proved the theorem, and this chapter does not either.
The source solves $e^x - 2x - 1 = 0$ on $[1,2]$ by iterating $g(x) = \ln(2x+1)$. It checks the hypotheses by hand: $g(1) = 1.0986\ldots$ and $g(2) = 1.6094\ldots$ lie in $[1,2]$; $g'(x) = 2/(2x+1)$ is decreasing, since $g'' = -4/(2x+1)^2 < 0$, so on $[1,2]$ it ranges over $[2/5, 2/3]$. By the mean value theorem $|g(x) - g(y)| \le \tfrac23 |x - y|$, so $L = 2/3$ and the iteration converges from any start in $[1,2]$. Starting from $x_0 = 1$ the source prints these iterates:
| k | xk | k | xk |
|---|---|---|---|
| 0 | 1.000000 | 6 | 1.245952 |
| 1 | 1.098612 | 7 | 1.250447 |
| 2 | 1.162283 | 8 | 1.253018 |
| 3 | 1.201339 | 9 | 1.254486 |
| 4 | 1.224563 | 10 | 1.255323 |
| 5 | 1.238121 | 11 | 1.255800 |
ch-rudolf-lipschitz-verify.py reproduces all twelve to the six printed decimals, using 60-digit
arithmetic, and checks the slope range exactly. It solves for $\xi$ to 50 digits, getting $\xi = 1.256431208626\ldots$,
and confirms the error bound $|x_k - \xi| \le (2/3)^k |x_0 - \xi|$ for $k = 0, \ldots, 40$. It also notes that
the observed ratio of successive errors is about $0.569$, which equals $g'(\xi)$: the bound $2/3$ is safe but the
iteration is faster than the bound says.
The source points out that a Lipschitz function need not be differentiable. $g(x) = |x|$ satisfies the condition with $L = 1$ on any closed interval and has slope $-1$ on one side of 0 and $+1$ on the other; the verifier prints exactly those two difference quotients. It adds, in a footnote, that a function satisfying the Lipschitz condition on an interval is differentiable except possibly on a set of Lebesgue measure zero, and names this Rademacher’s theorem. I did not check that statement.
The series has a Lean file, book5/ExistenceBanach.lean, that proves an existence-and-uniqueness
statement by exactly this route: the map is shown to be a contraction with constant $r < 1$, and Mathlib’s
ContractingWith and LipschitzWith supply the unique fixed point and the convergence of
the iterates. It works in a complete metric space instead of an interval, where completeness takes the place of
Brouwer’s theorem for existence; that generalisation is standard and is not in the source above. That file
has not been compiled by anyone yet. Nothing on this page is evidence that it compiles. The existing page
Book 5, Chapter V is a neighbour; I have not re-read it for this chapter.