Question

Let l be any positive integer. Use sandwich theorem for sequences to prove that the sequence {1/n^ l} converges to zero.

Answer 1

We consider the zero sequence, that is the sequence $(0)_{n=1}^{n=\infty}$

We have that for a positive integer $l, 0 \leq 1/n^l$ $.... *$

Also, we have that $n \leq n^l$

$\therefore 1/n^l \leq 1/n$ $.... **$

Combining * and **, we have $0 \leq 1/n^l \leq 1/n$

Since $\lim_{n \to \infty }0 = 0 = \lim_{n \to \infty} 1/n$

By sandwich theorem, we have that

$\lim_{n \to \infty} 1/n^l = 0$ as desired.