[tex]2 ,2+\frac{1}{2},2+\frac{1}{2+\frac{1}{2}}[/tex] 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 monotonicity:(by induction) [tex]a_1=2[/tex] [tex]a_2=2.5[/tex] so i guess its increasing suppose n=k is true: [tex]a_{k-1}<a_k[/tex] prove n=k+1 ([tex]a_{k}<a_{k+1}[/tex]) [tex] a_k>a_{k-1}\\ [/tex] [tex] \frac{1}{a_k}<\frac{1}{a_{k-1}}\\ [/tex] [tex] 2+\frac{1}{a_k}<2+\frac{1}{a_{k-1}}\\ [/tex] [tex] a_{k+1}<a_k [/tex] 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