$\displaystyle 2 ,2+\frac{1}{2},2+\frac{1}{2+\frac{1}{2}}$

etc..

(the sequence consists only from positive number so the sum is not negative)

in order to prove that its convergent i need to prove monotonicity and boundedness

monotonicityby induction)

$\displaystyle a_1=2$

$\displaystyle a_2=2.5$

so i guess its increasing

suppose n=k is true:

$\displaystyle a_{k-1}<a_k$

prove n=k+1 ($\displaystyle a_{k}<a_{k+1}$)

$\displaystyle

a_k>a_{k-1}\\

$

$\displaystyle

\frac{1}{a_k}<\frac{1}{a_{k-1}}\\

$

$\displaystyle

2+\frac{1}{a_k}<2+\frac{1}{a_{k-1}}\\

$

$\displaystyle

a_{k+1}<a_k

$

i proved the opposite

so this is weird.

the answer in the book tells me to split the sequence into odd /even sub sequences

the one is ascending and the other its descending.

i cant see how many sub sequences i need to split it to

maybe its 5 or 10

what is the general way of solving it.

and how you explained that i proved the opposite