proving the convergence of a sequence..

transgalactic

[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