The question goes...

*Let the sequences $\displaystyle \{a_{n}\}$ and $\displaystyle \{b_{n}\}$ be defined by $\displaystyle a_{1}=1, b_{1}=1$,*

*$\displaystyle a_{n+1}=a_{n}+\frac{1}{n(n+1)}$ and $\displaystyle b_{n+1}=b_{n}+\frac{1}{(n+1)^2}$.*

*Prove by induction that $\displaystyle a_{n}=2-\frac{1}{n}$ and $\displaystyle b_{n}\leq a_{n}$. What is $\displaystyle \lim_{n\rightarrow\infty}a_{n}$? Prove that $\displaystyle \lim_{n\rightarrow\infty}b_{n}$ exists.*

Can I have quite a detailed explanation please? I was never any good at proof by induction & I'm not too good at pure maths in general. Plus my lecturer's not been available for meetings lately