Question

Let {xn} n=1to ∞ and {yn}n=1to ∞ be sequences of positive numbers such that limn→∞ xn/yn = 0. Then show that if {yn}n=1to ∞ is bounded, then limn→∞ xn = 0

Answer 1

We have given that ${x_n}$ n=1 to ∞ and ${y_n}$ n=1to ∞ be sequences of positive numbers such that

$lim_{n\rightarrow\infty} \dfrac{x_n}{y_n}$

It is also given that $y_n$ is a bounded sequence

Hence $y_n$ is not equal to zero

therefore in the expression $lim_{n\rightarrow \infty }\dfrac{x_n}{y_n}$

$y_n \neq 0$

Therefore,

$lim_{n\rightarrow \infty}x_n = 0$