Well Taylor’s theorem is that any sufficiently differentiable function can be approximated by a particular polynomial plus a particular error. It just turns out that the error term doesn’t need to be very well behaved (eg consider the Taylor series at 0 for f(x)=exp(-1/x^2) ).

I think instead this is an application of the fundamental theorem of applied maths which states, approximately, that, in applied mathematics:

- all Taylor series converge

- all functions are piecewise smooth

- all sums and integrals can be transposed

- if the solution must be x if it exists, then the solution exists (and is x)

- if it looks right then it is