I am reading "Understanding Analysis" by Stephen Abbott.
I solved p.81 Exercise 2.8.3.
But I am not sure that my answer is correct.
Is the following answer correct or not?
Let $\{a_{ij} : i, j \in \mathbb{N}\}$ be a doubly indexed array of real numbers.
Suppose that $$ \sum_{i=1}^{\infty} \sum_{j=1}^{\infty} |a_{ij}| $$ converges.
Let $$ t_{mn} := \sum_{i=1}^{m} \sum_{j=1}^{n} |a_{ij}|. $$ Let $$ s_{mn} := \sum_{i=1}^{m} \sum_{j=1}^{n} a_{ij}. $$Exercise 2.8.3
(a) Prove that $(t_{nn})$ converges
(b) Now, use the fact that $(t_{nn})$ is a Cauchy sequence to argue that $(s_{nn})$ converges.
(a) For any $i \in \{1, 2, 3, \cdots \}$, $$ \sum_{j=1}^{n} |a_{ij}| \leq \sum_{j=1}^{\infty} |a_{ij}|. $$ So, $$ t_{nn} = \sum_{i=1}^{n} \sum_{j=1}^{n} |a_{ij}| \leq \sum_{i=1}^{n} \sum_{j=1}^{\infty} |a_{ij}| \leq \sum_{i=1}^{\infty} \sum_{j=1}^{\infty} |a_{ij}|. $$ So $(t_{nn})$ converges.
(b) $$| \sum_{i=1}^{n} \sum_{j=1}^{n} a_{ij} - \sum_{i=1}^{m} \sum_{j=1}^{m} a_{ij} | \leq | \sum_{i=1}^{n} \sum_{j=1}^{n} |a_{ij}| - \sum_{i=1}^{m} \sum_{j=1}^{m} |a_{ij}| |.$$
$(t_{nn})$ is a Cauchy sequence by assumption,
so $(s_{nn})$ is also a Cauchy sequence from the above inequality.