Textbook error relating to uncountability?

[JerryG]

I am reading a textbook about communications systems, and just came across a part that I believe to be wrong. As shown below, it says that if you take a vector g = [g(t1) g(t2) ... g(tN)] and take the limit as the number of samples goes to infinity, the result is a continuous function of t. Wouldn't this imply a bijection between the natural numbers and a continuous interval which was disproven by cantor's second theorem?

 

[Stephen Tashi]

Informal statements are often saved from outright errors by being vague. That passage did not give a definition for $lim_{N \rightarrow \infty} \mathbb{g}$.

The definition of a limit of a sequence of vectors (which can be regarded as a sequence of function) cannot be deduced from the definition of the type of limit used in calculus, which deals with the limit of a single function at a given point. There are actually many different types of limits that are defined for sequences of functions. I'd have to look these up and re-study them before I could swear that none of them make the book's statement true.

You are correct that the infinite sequence of discrete time samples does not ever become an uncountable set.

 

[pwsnafu]

There are actually many different types of limits that are defined for sequences of functions. I'd have to look these up and re-study them before I could swear that none of them make the book's statement true.

A "sequence of functions" would require the domains to be the same though...
The only option I could think is define the sequence of sequences
$h = [g(t_1), \ldots, g(t_N), 0, 0, \ldots]$
and then look at limits. Err.

The book really should have used step functions.

 

[Stephen Tashi]

A "sequence of functions" would require the domains to be the same though...
.

Yes, by many definitions it would. But statisticians say things like "For large N the binomial distribution can be approximated by a normal distribution". That's a statement connecting a function with a discrete domain to a function with a continuous domain. I don't know if people bother to formulate that type of approximation as a theorem about a limit of a sequence of functions, but I think if they wanted to, they could come up with something. There are so many possibilities for making up definitions!

 

[AlephZero]

This a nice example of the difference between math and engineering.

In engneering, any "real world" signal will be bandwith limited, and the statement in the book is true for bandwidth-limited signals.

But I agree that the book's "explanation" isn't much more than an arm-waving reason why that seems plausible.