For a continuous function $f : [0,1] \longrightarrow \mathbb{R}$, define $a_n(f) = \int_0^1 x^n f(x)\,\mathrm{d}x$. Choose the correct statement(s) from below: (A) The sequence $\{a_n(f)\}$ is bounded for every continuous function $f : [0,1] \longrightarrow \mathbb{R}$; (B) The sequence $\{a_n(f)\}$ is Cauchy for every continuous function $f : [0,1] \longrightarrow \mathbb{R}$; (C) The sequence $\{a_n(f)\}$ converges to 0 for every continuous function $f : [0,1] \longrightarrow \mathbb{R}$; (D) There exists a continuous function $f : [0,1] \longrightarrow \mathbb{R}$ such that the sequence $\{a_n(f)\}$ is divergent.
For a continuous function $f : [0,1] \longrightarrow \mathbb{R}$, define $a_n(f) = \int_0^1 x^n f(x)\,\mathrm{d}x$. Choose the correct statement(s) from below:\\
(A) The sequence $\{a_n(f)\}$ is bounded for every continuous function $f : [0,1] \longrightarrow \mathbb{R}$;\\
(B) The sequence $\{a_n(f)\}$ is Cauchy for every continuous function $f : [0,1] \longrightarrow \mathbb{R}$;\\
(C) The sequence $\{a_n(f)\}$ converges to 0 for every continuous function $f : [0,1] \longrightarrow \mathbb{R}$;\\
(D) There exists a continuous function $f : [0,1] \longrightarrow \mathbb{R}$ such that the sequence $\{a_n(f)\}$ is divergent.