Why shouldn’t they come together? Here’s a possible program. The first object to appear, in an elementary course of analysis, if one starts by an axiomatic presentation of $\mathbb R$, is the supremum. It is simple to understand visually, and people can be easily convinced that it is a powerful existence principle, and serves to define a variety of useful derived objects, that will be studied and used throughout the course. One can list them as a program for the course: the sum of a family of non-negative numbers, the limit inferior and superior of sequences, the uniform norm (introducing the useful notation $\|f\|_{\infty, S}$, for a function $f:S\to \mathbb R$), the length or variation of a curve, the Riemann-Darboux integral.

The definition of limit of a sequence comes naturally generalising the property of a supremum (infimum) of an increasing (decreasing) sequence. The notion of sum of nonnegative numbers extends to a absolutely summable families, and immediately allows interesting computations like $\sum_{n\ge0} a^n$, $\sum_{n\ge0} na^n$, $\sum_{n\ge1} 1/n$,.., once one proves convenient rules of calculus, which are a toy-version of measure theory: generalized associativity, monotone convergence, dominated convergence, Fubini-Tonelli. By dominated convergence, one immediately has a double definition of $e^x=\lim_{n\to\infty}(1+x/n)^n=\sum_{k=0}^\infty x^k/k!$, expanding the binomial, a computation that can be resumed for $x\in\mathbb C$, without typographical changes.