The following question is taken from here problem $2.5:$
Exercise $2.5:$ Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function such that $f(x) \geq 0$ for all $x$ and $$\int_{-\infty}^\infty f(x) \, dx = 1.$$ For $r\geq0,$ let $$I_n(x) = \int\dots\int_{x_1^2+x_2^2+\dots+x_n^2 \leq r^2} f(x_1) f(x_2)\dots f(x_n) \, dx_1 \, dx_2 \dots \, dx_n.$$ Find $\lim_{n\to\infty}I_n(x)$ for a fixed $r.$
The answer given is $0.$ I think I need to use $n$-dimensional spherical substitution to reduce the problem over an $n$-dimension sphere.
However, I have no idea how to use it. Any hint would be appreciated.