# Math Help - sequence, show convergence.

1. ## sequence, show convergence.

here it is:ImageShack - Image Hosting :: surehu5.jpg

for n >= 1 is decreasing, and positive for all n. Show {An} is convergent.

The series: $a_1=1$, $a_{n+1}=a_n\cdot\left(1-\frac {a_n}{2n}\right)$.
So we have to show that $0 and it is decreasing.
Using induction we prove that it's decreasing and positive. We know that $1=a_1>a_2=\frac 12$. Suppose that $a_{n-1}\cdot\left(1-\frac {a_{n-1}}{2n-2}\right)=a_n and $a_{n-1}>0$, $a_n>0$. We prove that: $a_n\cdot\left(1-\frac {a_n}{2n}\right)=a_{n+1} which is equivalent to: $0<\frac {a_n}{2n}$ by using the assumption $a_n>0$ and which is true. We only need to prove that $a_{n+1}>0$ or equivalently $\frac {a_n}{2n}<1\Leftrightarrow a_n<2n$. We know by assumption that $a_{n-1}<2n-2$ and because $a_n we're done $\Box$
So we know that $a_n$ is bounded and monotone therefore convergent.