Re: analysis with monotone convergence theorem.

In article <BgqGg.2676$0J6.1755@trnddc02>,
"TCL" <tlim1@xxxxxxxxxxx> wrote:

"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

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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.

It is right, but the cases A > 0, A < 0 should probably be dealt
with separately.
.