Lebl-Basic-Analysis-I · Problem 2.4.5

Sequences and Series

Mathematics · cauchy-sequences

Problem

Suppose a Cauchy sequence

\{x_n\}_{n=1}^\infty

is such that for every

M \in \mathbb{N}

, there exists a

k \geq M

and an

n \geq M

such that

x_k < 0

and

x_n > 0

. Using simply the definition of a Cauchy sequence and of a convergent sequence, show that the sequence converges to 0.