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# 数学代写|实分析代写Real Analysis代考|Interchange of Limits

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## 数学代写|实分析代写Real Analysis代考|Interchange of Limits

Let $\left{b_{i j}\right}$ be a doubly indexed sequence of real numbers. It is natural to ask for the extent to which
$$\lim i \lim _j b{i j}=\lim j \lim _i b{i j}$$
more specifically to ask how to tell, in an expression involving iterated limits, whether we can interchange the order of the two limit operations. We can view matters conveniently in terms of an infinite matrix
$$\left(\begin{array}{ccc} b_{11} & b_{12} & \cdots \ b_{21} & b_{22} & \ \vdots & & \ddots \end{array}\right)$$
The left-hand iterated limit, namely $\lim i \lim _j b{i j}$, is obtained by forming the limit of each row, assembling the results, and then taking the limit of the row limits down through the rows. The right-hand iterated limit, namely $\lim j \lim _i b{i j}$, is obtained by forming the limit of each column, assembling the results, and then taking the limit of the column limits through the columns. If we use the particular infinite matrix
$$\left(\begin{array}{ccccc} 1 & 1 & 1 & 1 & \cdots \ 0 & 1 & 1 & 1 & \cdots \ 0 & 0 & 1 & 1 & \cdots \ 0 & 0 & 0 & 1 & \cdots \ \vdots & & & & \ddots \end{array}\right)$$
then we see that the first iterated limit depends only on the part of the matrix above the main diagonal, while the second iterated limit depends only on the part of the matrix below the main diagonal. Thus the two iterated limits in general have no reason at all to be related. In the specific matrix that we have just considered, they are 1 and 0 , respectively. Let us consider some examples along the same lines but with an analytic flavor.

## 数学代写|实分析代写Real Analysis代考|Uniform Convergence

Let us examine more closely what is happening in the proof of Theorem 1.13 , in which it is proved that iterated limits can be interchanged under certain hypotheses of monotonicity. One of the iterated limits is $L=\lim i \lim _j b{i j}$, and the claim is that $L$ is approached as $i$ and $j$ tend to infinity jointly. In terms of a matrix whose entries are the various $b_{i j}$ ‘s, the pictorial assertion is that all the terms far down and to the right are close to $L$ :
$$\left(\begin{array}{cc} \cdots & \cdots \ \cdots & \begin{array}{l} \text { All terms here } \ \text { are close to } L \end{array} \end{array}\right) \text {. }$$
To see this claim, let us choose a row limit $L_{i_0}$ that is close to $L$ and then take an entry $b_{i_0 j_0}$ that is close to $L_{i_0}$. Then $b_{i_0 j_0}$ is close to $L$, and all terms down and to the right from there are even closer because of the hypothesis of monotonicity.
To relate this behavior to something uniform, suppose that $L<+\infty$, and let some $\epsilon>0$ be given. We have just seen that we can arrange to have $\left|L-b_{i j}\right|<\epsilon$ whenever $i \geq i_0$ and $j \geq j_0$. Then $\left|L_i-b_{i j}\right|<\epsilon$ whenever $i \geq i_0$, provided $j \geq j_0$. Also, we have $\lim j b{i j}=L_i$ for $i=1,2, \ldots, i_0-1$. Thus $\left|L_i-b_{i j}\right|<\epsilon$ for all $i$, provided $j \geq j_0^{\prime}$, where $j_0^{\prime}$ is some larger index than $j_0$. This is the notion of uniform convergence that we shall define precisely in a moment: an expression with a parameter ( $j$ in our case) has a limit (on the variable $i$ in our case) with an estimate independent of the parameter. We can visualize matters as in the following matrix:
$$i\left(\begin{array}{l|l} j & j_0^{\prime} \ \cdots & \begin{array}{l} \text { All terms here } \ \text { are close to } L_i \ \text { on all rows. } \end{array} \end{array}\right) .$$
The vertical dividing line occurs when the column index $j$ is equal to $j_0^{\prime}$, and all terms to the right of this line are close to their respective row limits $L_i$.

## 数学代写|实分析代写Real Analysis代考|Interchange of Limits

$$\lim i \lim j b{i j}=\lim j \lim _i b{i j}$$ 更具体地说，是问如何判断，在一个包含迭代极限的表达式中，我们是否可以交换两个极限运算的顺序。我们可以方便地用无穷矩阵的形式来考虑问题 $$\left(\begin{array}{ccc} b{11} & b_{12} & \cdots \ b_{21} & b_{22} & \ \vdots & & \ddots \end{array}\right)$$

$$\left(\begin{array}{ccccc} 1 & 1 & 1 & 1 & \cdots \ 0 & 1 & 1 & 1 & \cdots \ 0 & 0 & 1 & 1 & \cdots \ 0 & 0 & 0 & 1 & \cdots \ \vdots & & & & \ddots \end{array}\right)$$

## 数学代写|实分析代写Real Analysis代考|Uniform Convergence

$$\left(\begin{array}{cc} \cdots & \cdots \ \cdots & \begin{array}{l} \text { All terms here } \ \text { are close to } L \end{array} \end{array}\right) \text {. }$$

$$i\left(\begin{array}{l|l} j & j_0^{\prime} \ \cdots & \begin{array}{l} \text { All terms here } \ \text { are close to } L_i \ \text { on all rows. } \end{array} \end{array}\right) .$$

## MATLAB代写

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