Graphics Reference
In-Depth Information
A
for all
n
(a) Show that
a
, by contradiction, supposing that
A
a
A
, for some positive integer
k
, and showing that
the subsequence (beyond the
n
th term) could not then be
both
monotonic increasing
and
convergent to
A
.
(b) Use the fact that every term of the sequence (
a
) is followed
eventually by a term of the convergent subsequence to prove
that
a
A
for all
n
.
(c) For any positive
, use the convergence of the subsequence to
A
to find a term of the subsequence between
A
and
A
.
(d) Namean
N
such that
n
N
a
A
, so that (
a
)
A
.
State the theorem you have proved through (a), (b), (c) and (d). State an
analogous theorem for monotonic decreasing sequences. Combine the
statements of these two theorems into a single theorem about the
convergence of monotonic sequences.
Intuition and convergence
81 An isosceles right-angled triangle is constructed with a given line
segment
l
as hypotenuse. It has perimeter
p
.
Two
isosceles right-angled triangles are constructed each with
hypotenuse on half of the line segment
l
, but with thetwo
hypotenuses occupying the whole of
l
. Thetwo triangls, which
only overlap at one vertex, have combined perimeter
p
and area
A
and
combined area
A
.