Proving the convergence of a sequence (1 Viewer)

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[tex]2 ,2+\frac{1}{2},2+\frac{1}{2+\frac{1}{2}}[/tex]
(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)

so i guess its increasing
suppose n=k is true:
prove n=k+1 ([tex]a_{k}<a_{k+1}[/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
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