completeness of Lp
Completeness is the property of having no missing limits. A space is complete if every sequence that ‘ought’ to converge — because its terms eventually huddle arbitrarily close together — really does converge to a point of the space. The rationals fail this test (the truncations of pi cluster but their limit is irrational); the reals pass it. Completeness of L^p says the same good behavior holds for whole functions measured in the L^p norm.
Concretely: in L^p(X, mu) with 1 <= p <= infinity, if a sequence f_n satisfies ||f_n - f_m||_p -> 0 as n, m -> infinity (a Cauchy sequence), then there is a function f in L^p with ||f_n - f||_p -> 0. This is precisely the content of the Riesz–Fischer theorem, and it is what upgrades the merely normed space L^p to a Banach space. For p = 2 it makes L^2 a Hilbert space.
Why it matters: completeness is what lets analysis proceed by approximation and limiting arguments. You construct a solution as a limit of approximations, verify the approximations form a Cauchy sequence, and completeness hands you the limit for free — guaranteed to be a genuine L^p function. Fixed-point theorems, existence proofs for differential and integral equations, and the convergence of Fourier expansions all lean on it.
Truncations of a function illustrate completeness as a feature, not a bug: on [0, 1] let g_n agree with x^(-1/4) where that is at most n, and equal n otherwise. Each g_n is bounded, hence in L^2; they form a Cauchy sequence in L^2 and converge to x^(-1/4), which is in L^2 — the limit did not escape the space.
Cauchy sequences of nice functions can converge to a less nice — but still L^p — limit.