Problem 1

Let $\{x_n\}$ be a sequence of positive real numbers such that $x_n^3>n.$ Show that $1/x_n\to0$ as $n\to\infty$ with respect to the usual metric on $\mathbb{R}.$

Proof. [Solution] The sequence $y_n=1/x_n$ satifies $0<y^3_n<1/n.$ Given $\epsilon>0$ choose $N>1/\epsilon^3,$ then $n>N$ implies $y_n^3<1/n<1/N<\epsilon ^3$ so $y_n<\epsilon$ and it follows that $y_n\to0$ as $n\to\infty.$ $\qedsymbol$