请教数学分析习题

kindlychung

见附件图片。谢谢。

elimqiu

kindlychung

[这个贴子最后由kindlychung在 2010/09/23 09:35am 第 1 次编辑]

谢谢。
未名BBS上有人给出另一个解法：

根据mn顺序的不同，计算结果完全不一样，很费解。麻烦指点，谢谢。

elimqiu

比较法的极限形式的证明，就是我的证法。
please post Definition 2.2.3
What book is this?

kindlychung

哦，原来是你回答的啊，巧了。谢谢你。我很羡慕懂数学的人。
附件是我现在看的书。

kindlychung

pdf文件可能超过容量上限了，发个图片吧:
Understanding Analysis

elimqiu

所以两种方式都能刻划 (m,n) 的 大小（模/范数）

kindlychung

[这个贴子最后由kindlychung在 2010/09/24 02:30pm 第 1 次编辑]

Thank you for your patience and kindness.
However large m+n may be, m/(m+n) still varies between 0 and 1, as a matter of fact, the interval of variation gets even larger when m+n tends to infinity. So, according to your definition above, how can it converge to any L?
And please forgive me, I really don';t get the point of the last formula (the inequality involving m+n).

elimqiu

The given array {a(m,n)} is divergent by the definition. You are right: there is no constant L can fit the definition.
What we can see is that double limit exists → iterated limits exist and equal
but the reverse is not true in general. That means double limit requires much more than iterated limits.
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The inequality shows that using m+n or (m^2+n^2)^(1/2) for the way of measure a vector';s "length" have no differences regard to limit. In topology, these two of metric definition produce the same 2D topology.

kindlychung

Here is another one.
x is in (A cap B) implies x is in A and x is in B, which in turn implies that f(x) is in f(A) and f(x) is f(B), hence f(x) is in (f(A) cap f(B)).
Therefore, f(x) in f(A cap B) => f(x) in (f(A) cap f(B)).
The rest of the exercise baffles me, please help.