successive approximations
Successive approximations is the general strategy of solving a hard problem by producing a sequence of ever-better guesses, each computed from the one before, and arguing that the sequence settles onto the true answer. In differential equations it is the same engine as Picard iteration — indeed the two names are used interchangeably — but the idea is broader and worth seeing on its own terms, because the pattern recurs all over mathematics.
The mechanism is always the same shape. You recast the problem as 'find a thing x with x = T(x)' for some operation T that takes a candidate and returns a refined candidate. Then you iterate: x1 = T(x0), x2 = T(x1), x3 = T(x2), and so on, hoping the xn home in on a fixed point where applying T changes nothing. For initial value problems the candidate is a function and T is the integral operator that sends y to y0 plus the integral of f(t, y(t)) dt; for a square root by Newton's method the candidate is a number; for solving a linear system iteratively the candidate is a vector. Convergence is guaranteed when T is a contraction — when each pass shrinks the distance to the answer by a fixed factor — and then the errors fall off geometrically.
The honest point is what makes it work and when it doesn't. The whole method rides on T being a contraction; without that shrinking property, the guesses can drift, oscillate, or run away rather than converge. That is precisely why the Lipschitz condition matters in the ODE setting — it is exactly the hypothesis that makes the relevant operator a contraction on a short enough interval. Successive approximations is thus less a trick than a unifying lens: many existence proofs and many numerical schemes are the same move, build a contraction and iterate.
To find a root of x = cos(x), iterate x_{n+1} = cos(x_n) from any start, say x0 = 1: you get 0.540, 0.858, 0.654, 0.793, ... slowly closing in on about 0.739. Cosine is a contraction on this range, so the successive approximations converge to the fixed point.
Same idea as Picard, stripped to its bones: a contraction iterated from any guess converges to the fixed point.
Iterating only converges if the map genuinely shrinks distances. Iterate a non-contraction (say x_{n+1} = 2x_n) and the guesses fly apart instead of settling — the contraction property is doing all the work.