This talk is designed as a bit of a mathematical magic show: we will live demo Julia packages that allow the construction of infinite-dimensional vectors and matrices, infinite matrix factorisation, solving infinite-dimensional linear systems, and even computation of spectrum. This methodology is also applicable to solving linear ordinary and partial differential equations and singular integral equations. The “grand reveal” is that many of the techniques also work with interval arithmetic, almost for free. One can, for example, compute interval bounds on zeros of Bessel functions by simply specifying the 3-term recurrence they satisfy. Peeling back the curtains, we will describe how underlying structure in the operators makes this possible, but also remaining gaps that mean that the leap to interval arithmetic is not quite as rigorous as it appears to be, and can in certain cases fail spectacularly.