Actuarial Mathematics proof error?

Question

Is there something wrong with this proof or is it just me? Did they forget the -ve sign or does it cancel somehow? (From the second to last to last line)

Answer 1

Remember that the survival function (and consequently the deterministic number of lives) is a nonincreasing function with $s(\infty) = l_\infty = 0$. Consequently, by the fundamental theorem, $$\int_{y=x}^\infty l_y \mu(y) \, dy = g(\infty) - g(x),$$ where $g$ is an antiderivative of the integrand, i.e., $g$ satisfies $g'(y) = l_y \mu(y)$. But since $g(\infty) = 0$, upon differentiation, we obtain $$\frac{dl_x}{dx} = -g'(x) = -l_x \mu(x),$$ as desired.

Answer 2

Because the lower limit of the integral is $x$, it is correct to change signs. Informally, you can check by differentiating, using the Fundamental Theorem of Calculus.