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