# True or false if $a_n$ is a decreasing sequence of positive numbers$b_n$ converges to 0, then $a_n\over b_n$ diverges.

True or false: if $a_n$ is any decreasing sequence of positive real numbers and $b_n$ is any sequence of real numbers converges to 0, then $a_n\over b_n$ diverges.

if $a_n=e^{-n}$ and $b_n=\frac {1}{n^2}$ then

$(a_n)$ is a decreasing sequence of positive numbers and $\lim_{+\infty}b_n=0$

but

$$\frac {a_n}{b_n}=n^2e^{-n}\to 0.$$

Hint: Take $a_n=b_n=\frac{1}{n}$

• ya, that's what I did, but I wasn't sure if it was correct. A sequence of decreasing positive numbers should converge to zero. thanks – Jasmine Jul 24 '17 at 0:56
• @Jasmine "A sequence of decreasing positive numbers should converge to zero"... that statement is not at all true. $c_n=1+\frac{1}{n}$ is a decreasing sequence of positive real numbers which converges to $1$ instead. It just so happens that the example tattwamasi gives converges to zero in this specific case. – JMoravitz Jul 24 '17 at 0:58
• @Jasmine: A sequence of decreasing positive numbers need not converge to zero. $c_n=1+\frac 1n$ is decreasing and positive but converges to $1$. This example meets your requirements on $a_n,b_n$ and it is constant so it converges. – Ross Millikan Jul 24 '17 at 0:59
• thanx for the correction – Jasmine Jul 24 '17 at 1:24

Hint: Take $a_n = b_n$ for all $n$