Problem. Let $(f_n)_{n=1}^\infty$ be a sequence of functions $f_n\colon [-1,\infty)^n\to\mathbb{R}$ that are recursively defined in the following way:
$f_1(x_1)=1+x_1,$
$f_n(x_1,\ldots,x_n) = \begin{cases} (1+x_n)f_{n-1}(x_1,\ldots,x_{n-1}) & \text{if } f_{n-1}(x_1,\ldots,x_{n-1}) \le 1, \\ x_n + f_{n-1}(x_1,\ldots,x_{n-1}) & \text{otherwise}. \end{cases}$
Let $(g_n)_{n=1}^\infty$ be a sequence of functions $g_n\colon [-1,\infty)^n\to\mathbb{R}$ that are defined in the following way:
$g_n(x_1,\ldots,x_n) = \prod_{k=1}^n (1+x_k).$
Show, for any $n\ge1$ and any $x_1,\ldots,x_n\in[-1,\infty)$, that if $f_n(x_1,\ldots,x_n) <1$, it also holds that $g_n(x_1,\ldots,x_n) \le1$.
Observations. I have tried doing this with induction, but that doesn’t seem to work. It appears that I need more intricate knowledge of the relationship between $f_n$ and $g_n$ in order to solve the problem. Clearly, both are multivariate polynomials. It also seems like all the terms in $f_n$ are also contained in $g_n$, but I don’t know how that could help.
Real-life motivation. Note that the quantity $g_n(x_1,\ldots,x_n)$ is nothing but 1 dollar that has been invested in $n$ different ventures with percentage returns (positive or negative) of $x_1,\ldots,x_n$, where the profits from each venture have been reinvested in the next venture. The quantity $f_n(x_1,\ldots,x_n)$ is similar, except that profits are not reinvested. The objective is to show that if investing in ventures without reinvesting profits results in a loss, then reinvesting the profits couldn’t possibly have resulted in a profit.