Assuming that there are uncountably infinitely many irrationals between any two consecutive rationals, how can the Dedekind cuts (defined on the countably infinite rationals) define all the irrationals?
For instance, is it not true that the real interval between 1 and the largest element of the Dedekind cut with all elements < 1 (i.e. largest element of the cut is $\lim_{n\to\infty} {n\over n+1}$ , natural n) contains uncountably infinitely many irrationals?
You have several misconceptions about the way numbers work; all of these are common.
First you seem to have the idea that the rationals are a discrete sequence, like the integers, so that every rational number has a predecessor rational and a successor rational. This is not the case. For suppose you claim that $a$ and $b$ are consecutive rationals. But $a\lt\frac{a+b}{2}\lt b,$ and $\frac{a+b}{2}$ is rational, so we have just found a rational number between $a$ and $b$, contradicting your claim that $a$ and $b$ were consecutive. But this construction works for any rationals $a$ and $b$. So there is no such thing as consecutive rationals.
Second, you seem to think that $$ \lim_{n\to \infty}\frac{n}{n+1} $$ means something other than what it does. This notation is a mathematical term of art with a particular, fixed meaning. Without going into details, the $\lim$ notation describes a property that a number might or might not have, and in this particular case one can show that the one and only number with that particular property is the number 1. It is not some number that is magically less than 1 by a tiny amount; it is equal to 1. When you write $\lim_{n\to \infty}\frac{n}{n+1}$, you are intending to communicate a certain idea, but what you have actually written is just a complicated notation for the number 1.
What I believe you are trying to do here is to invent a number $y$ so that $y$ is less than 1, but still closer to 1 than any other number less than 1. This is a common desire, but there is no such $y$. There can't be, because $y$ has contradictory properties. Consider $e = 1-y$. This is positive, because $y<1$. Then $0 < \frac e 2 < e$, so $$ y < y + \frac e2 < y+e = 1 $$ which shows that $y+\frac e2$, while still less than 1, is even closer to 1 than $y$ was, contradicting the original definition of $y$. This shows that the idea for $y$ is incoherent; there is no such object with that property.
It is a common error in mathematics (and philosophy!) to assume that just because some set of properties can be stated, that there must be some object with those properties. This is not the case. One can say
“the current Crown Prince of the Ottoman Empire”,
but there is at present no Ottoman Empire and no Crown Prince of the Ottoman Empire, so the phrase, although meaningful, does not refer to any actual thing, although you have to know a bit about the Ottoman Empire to know this. Similarly one can say
“the rational number preceding 1”
or
“the largest real number less than 1”
and it seems at first to make sense. But if you know a bit more about numbers you know that those phrases do not designate any actual object.
I'm not sure this answers your question, but your question is based on several basic misunderstandings of how numbers work, so I hope this helps clear up some of it.
