Re: analysis with monotone convergence theorem.
- From: "TCL" <tlim1@xxxxxxxxxxx>
- Date: Mon, 21 Aug 2006 22:16:01 GMT
"mina_world" <mina_world@xxxxxxxxxxx> wrote in message
news:eccpvk$8tb$1@xxxxxxxxxxxxxxxxxxx
hello sir~
sequence {x_n} is bounded.
2(x_n) <= x_(n-1) + x_(n+1)
show that lim{n->00} {x_n - x_(n-1)} = 0
-----------------------------------------------
i can do it.
2(x_n) <= x_(n-1) + x_(n+1)
=> x_n - x_(n-1) <= x_(n+1) - x_n
so, new sequence {x_n - x_(n_1)} is monotone increasing.
of course, this sequence is bounded.
thus {x_n - x_(n_1)} converges by monotone convergence
theorem.
and i must show that lim{n->00} {x_n - x_(n-1)} = 0.
suppose that lim {x_n - x_(n-1)} = A (A =/= 0).
so,
there exists N such that n>= N => |x_n - x_(n-1)| > |A|/2.
so,
|x_(N+1) - x_N| > |A|/2
and
|x_(N+2) - x_N| = |x_(N+2) - x_(N+1) + x_(N+1) - x_N| > 2*(|A|/2)
Wait a minute. The last inequality above is NOT right.
...........
|x_(N+n) - x_N| > n*(|A|/2)
so, |x_(N+n) - x_N| -> 00 as n->00.
but since x_n is bouned, |x_(N+n) - x_N| is bounded.
thus contradiction.
thus lim{n->00} {x_n - x_(n-1)} = 0.
-----------------------------------------------------
is this no problem ?
if there is a easy solution, i hope to know that from you.
so, i need your advice.
.
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