A summable numerical majorant gives uniform and absolute convergence of a series of functions. The point is to make the formal expression readable enough to audit line by line.
Start locally
A sequence \(f_n:X\to\mathbf R\) converges uniformly to \(f\) when \(\sup_{x\in X}|f_n(x)-f(x)|\to0\). The supremum norm captures one error bound valid everywhere.
The definition determines which expressions are legal; only then does the identity become meaningful. An equality in \(\mathcal A\) may change ambient meaning, so I keep \(\mathsf D\) separate from \(\mathsf C\).
Compute before generalising
The middle display is intentionally dense: it is where signs, bounds, multiplicities, or normalising factors are most likely to be lost.
The global view
The invariant statement is the one that does not depend on a convenient choice of coordinates, representatives, basis, or enumeration.
Edge conditions
Pointwise convergence allows the bad point to move with \(n\). Continuity, integration, and differentiation survive limits under different uniform hypotheses.
The result is compact enough to reuse without pretending that the caveat has disappeared. The worked line remains the quickest consistency check.