Consider the two positive integer sequences, defined for a fixed positive integer $c \geq 2$
\[f(n)=\frac{1}{n}\left\lfloor\frac{n}{c}\right\rfloor, \quad g(n)=n\left\lfloor\frac{c}{n}\right\rfloor\]
where $\lfloor t\rfloor$ denotes the largest integer with value at most $t$. Which of the following statements is $\text{TRUE}$ as $n \rightarrow \infty$ ?
- Both sequences converge to zero
- The first sequence does not converge, while the second sequence converges to $0$
- The first sequence converges to zero, while the second sequence does not converge
- The first sequence converges to $1 / c$, while the second sequence converges to $0$
- The first sequence converges to $1 / c$, while the second sequence converges to $c$