Let $x_k$ be real, for $k$ in some index set $K.$
Let $e_n$ be the $n$th-degree elementary symmetric function of $x_k,\,\,k\in K,$ i.e. $$ e_n = \sum_{\left\{ \begin{array}c L\,\subseteq\,K \\ |L|\,=\, n \end{array} \right\}} \prod_{k\,\in\, L} x_k. $$
Consider the following argument \begin{align} & e_0+e_1 + e_2 + e_3 + \cdots \\[10pt] = {} & 1 + (x_1+x_2+x_3+\cdots) \\ & \qquad {} + (x_1x_2 + x_1x_3 + x_2x_3+ \cdots) + (x_1x_2x_3+\cdots) + \cdots \\[10pt] = {} & (1+x_1)(1+x_2)(1+x_3) \cdots \\[10pt] \le {} & \exp(x_1)\exp(x_2)\exp(x_3) \cdots \\[10pt] = {} & \exp(x_1+x_2+x_3+\cdots). \end{align} If the index set $K$ is finite, then this argument is a valid proof of an inequality.
If $K$ is infinite, then we might suppose that the series $\sum_{k\in K} x_k$ converges absolutely. In that case I would like this argument to be used in showing $\sum_{n\ge0} e_n$ also converges absolutely. That can be done fairly routinely, but I'm wondering if there's a way to condense everything about convergence and limits and analysis into a one pithy line, where the reader might be expected to fill in the details, but they would be implicitly present in that one line. And has that been published somewhere? (I want to use it in something where the principal focus would be on something other than how this is proved, and the argument would be just a sort of reminder of something you learned at your mother's knee.)
Alright; I've realized that what I'd already written will do it if I just make the language more severely terse. It will say that if $f_n$ is the $n$th-degree elementary symmetric function of $y_k = |x_k|,\,\,\,k\in K,$ then the series in the question is dominated by \begin{align} & f_0+f_1 + f_2 + f_3 + \cdots \\[10pt] = {} & 1 + (y_1+y_2+y_3+\cdots) \\ & \qquad {} + (y_1y_2 + y_1y_3 + y_2y_3+ \cdots) + (y_1y_2y_3+\cdots) + \cdots \\[10pt] = {} & (1+y_1)(1+y_2)(1+y_3) \cdots \\[10pt] \le {} & \exp(y_1)\exp(y_2)\exp(y_3) \cdots \\[10pt] = {} & \exp(y_1+y_2+y_3+\cdots). \end{align} Since everything here is nonnegative, the reader will live happily ever after.
I will just consider the phrase "is dominated by" to take care of all the analysis.