By definition, the random variables $X_1,\dots,X_n$ are independent iff $$ \Pr(X_1\in B_1,\dots,X_n\in B_n) = \Pr(X_1\in B_1)\dots\Pr(X_n\in B_n) $$ for every choice of Borel sets $B_1,\dots,B_n$. Hence, picking $B_i=(-\infty,t_i]$, we have $$ \Pr(X_1\leq t_1,\dots,X_n\leq t_n) = \Pr(X_1\leq t_1)\dots\Pr(X_n\leq t_n). \qquad (*) $$ If each $X_i$ has a density $f_{X_i}$, then the RHS of $(*)$ is equal to $$ \left(\int_{-\infty}^{t_1} f_{X_1}(x_1)\,dx_1\right) \dots \left(\int_{-\infty}^{t_n} f_{X_n}(x_n)\,dx_n\right). $$ By Fubini's theorem, this is equal to $$ \int_{-\infty}^{t_1}\dots\int_{-\infty}^{t_n} f_{X_1}(x_1)\dots f_{X_n}(x_n)\,dx_1\dots dx_n. $$ Hence, it follows that the random vector $(X_1,\dots,X_n)$ has density $$ f_{X_1,\dots X_n}(x_1,\dots,x_n) = f_{X_1}(x_1)\dots f_{X_n}(x_n). $$

By definition, the random variables $X_1,\dots,X_n$ are independent iff $$ \Pr(X_1\in B_1,\dots,X_n\in B_n) = \Pr(X_1\in B_1)\dots\Pr(X_n\in B_n) $$ for every choice of Borel sets $B_1,\dots,B_n$. Hence, picking $B_i=(-\infty,t_i]$, we have $$ \Pr(X_1\leq t_1,\dots,X_n\leq t_n) = \Pr(X_1\leq t_1)\dots\Pr(X_n\leq t_n). \qquad (*) $$ If each $X_i$ has a density $f_{X_i}$, then the RHS of $(*)$ is equal to $$ \left(\int_{-\infty}^{t_1} f_{X_1}(x_1)\,dx_1\right) \dots \left(\int_{-\infty}^{t_n} f_{X_n}(x_n)\,dx_n\right). $$ By Fubini's theorem, this is equal to $$ \int_{-\infty}^{t_n}\dots\int_{-\infty}^{t_1} f_{X_1}(x_1)\dots f_{X_n}(x_n)\,dx_1\dots dx_n. $$ Hence, it follows that the random vector $(X_1,\dots,X_n)$ has density $$ f_{X_1,\dots X_n}(x_1,\dots,x_n) = f_{X_1}(x_1)\dots f_{X_n}(x_n). $$

By definition, the random variables $X_1,\dots,X_n$ are independent iff $$ \Pr(X_1\in B_1,\dots,X_n\in B_n) = \Pr(X_1\in B_1)\dots\Pr(X_n\in B_n) $$ for every choice of Borel sets $B_1,\dots,B_n$. Hence, picking $B_i=(-\infty,x_i]$, we have $$ \Pr(X_1\leq x_1,\dots,X_n\leq x_n) = \Pr(X_1\leq x_1)\dots\Pr(X_n\leq x_n). \qquad (*) $$ If each $X_i$ has a density $f_{X_i}$, then the RHS of $(*)$ is equal to $$ \left(\int_{-\infty}^{x_1} f_{X_1}(t_1)\,dt_1\right) \dots \left(\int_{-\infty}^{x_n} f_{X_n}(t_n)\,dt_n\right). $$ By Fubini's theorem, this is equal to $$ \int_{-\infty}^{x_1}\dots\int_{-\infty}^{x_n} f_{X_1}(t_1)\dots f_{X_n}(t_n)\,dt_1\dots dt_n. $$ Hence, it follows that the random vector $(X_1,\dots,X_n)$ has density $$ f_{X_1,\dots X_n}(x_1,\dots,x_n) = f_{X_1}(x_1)\dots f_{X_n}(x_n). $$

By definition, the random variables $X_1,\dots,X_n$ are independent iff $$ \Pr(X_1\in B_1,\dots,X_n\in B_n) = \Pr(X_1\in B_1)\dots\Pr(X_n\in B_n) $$ for every choice of Borel sets $B_1,\dots,B_n$. Hence, picking $B_i=(-\infty,t_i]$, we have $$ \Pr(X_1\leq t_1,\dots,X_n\leq t_n) = \Pr(X_1\leq t_1)\dots\Pr(X_n\leq t_n). \qquad (*) $$ If each $X_i$ has a density $f_{X_i}$, then the RHS of $(*)$ is equal to $$ \left(\int_{-\infty}^{t_1} f_{X_1}(x_1)\,dx_1\right) \dots \left(\int_{-\infty}^{t_n} f_{X_n}(x_n)\,dx_n\right). $$ By Fubini's theorem, this is equal to $$ \int_{-\infty}^{t_1}\dots\int_{-\infty}^{t_n} f_{X_1}(x_1)\dots f_{X_n}(x_n)\,dx_1\dots dx_n. $$ Hence, it follows that the random vector $(X_1,\dots,X_n)$ has density $$ f_{X_1,\dots X_n}(x_1,\dots,x_n) = f_{X_1}(x_1)\dots f_{X_n}(x_n). $$