Could you please write the formulas in the forum using LaTeX in the future as probably very few people will check your handwriting on ImageShack...
The series: , .
So we have to show that and it is decreasing.
Using induction we prove that it's decreasing and positive. We know that . Suppose that and , . We prove that: which is equivalent to: by using the assumption and which is true. We only need to prove that or equivalently . We know by assumption that and because we're done
So we know that is bounded and monotone therefore convergent.