Htykuut

Lots of textbooks on Mathematical Analysis say that:

Cauchy's Convergence Theorem shows that, any convergent sequence
consisting of real numbers has a real number limit, which is called the completeness of real numbers.

In my opinion, the completeness of real numbers refers to that, if a real number sequence is convergent, it is necessarily convergent to a real number. By contrast, a rational number sequence is not necessarily convergent to a rational number,even though it converges. A classical example is that $left(1+dfrac{1}{n}right)^n to e(n to infty).$ In another word, the rational number set does not have the completeness property.

But how does the completeness of real numbers come out from Cauchy's Theorem? How to comprehend the point?

There is a subtlety here regarding convergence that perhaps you have missed. When you say that

In my opinion, the completeness of real numbers refers to that, if a real number sequence is convergent, it is necessarily convergent to a real number.

you are making a slight (but crucial) error.

Firstly, it is improper to suggest that a sequence in a space $X$ converges, but its limit lies outside $X$. Because, if a sequence is convergent then by definition its limit must lie in the same space.

However, there are sequences that appear to converge: their tails lie in intervals of smaller and smaller length, tending to zero. Such sequences are called Cauchy sequences.

Now, a Cauchy sequence in a space $X$ may or may not have a limit in the space $X$. For instance, there are Cauchy sequences in $mathbb{Q}$, such as the one you have chosen,
$$
a_n = left( 1 + frac1n right)^{n},
$$
which do not converge in $mathbb{Q}$.

The important theorem here is that every Cauchy sequence in $mathbb{R}$ converges. This is perhaps what you meant by the Cauchy Convergence Theorem? Note the difference between the above statement and the one you have quoted in your post.

So, if you want to know why the Cauchy Convergence Theorem implies that $mathbb{R}$ is complete, it is because completeness is defined that way: a space is said to be complete if every Cauchy sequence in $X$ is convergent. And the Cauchy Convergence Theorem says that $mathbb{R}$ is complete.

This might be unsatisfactory, because it might appear that all we have done is make a clever definition. But I'll be happy to address any concerns if you have in the comments.

well, if you mean by cauchy convergence theorem that any cauchy sequence converges, then yes, because this is precisely the definition of being complete and $mathbb{R}$ often gets defined as the completion of $mathbb{Q}$. However, your second statement is wrong, since convergence demands a limit point! hence the sequence you wrote down does not converge in $mathbb{Q}$, but it defines a cauchy sequence there! The existence of the limit is essential! so please revise your definition of convergence.