Is $0\bar{.9}$ in $[0,1)$?

Question

Is $0\bar{.9}$ in $[0,1)$?

$0\bar{.9} = 1$, so does that mean $0\bar{.9} \notin [0,1)$?

What about

$$\lim_{n\to \infty} 1/n^2 = 0$$ is this in $(0,1]$?

Or are these two separate concepts? If so, please help me understand them.

Answer 1

A sequence is not its limit. Sequences need not even have a limit.

But even a convergent sequence is not its limit. It's true, intuitively we think about continuity everywhere (until this wrong intuition is uprooted by the myriad of examples).

Since you are thinking about $0.\overline 9$ as the limit of $0.9, 0.99, 0.999,\dots$ and all of these are clearly are members of $[0,1)$, but the limit need not be in the set. Not without additional hypotheses (e.g. closure).

The answer is similar about $0$ in $(0,1]$.

What is true, is that $1$ is a limit point of $[0,1)$, exactly because it is in the limit of the sequence $0.9,0.99,0.999,\ldots$; and similarly $0$ is a limit point of $(0,1]$ since it is the limit of $\frac12,\frac13,\frac14,\ldots$

But as neither $[0,1)$ nor $(0,1]$ are closed, there is no reason for them to contain all their limit points.

Answer 2

If two things are equal, that means they are literally exactly the same thing in every possible way. So, $0.\bar{9} = 1$ means that $0.\bar{9}$ and $1$ are identical: they're just two different sequence of symbols we can use to refer to the same number. So, since $1\not\in[0,1)$, $0.\bar{9}\not\in[0,1)$ as well. Similarly, $\lim_{n\to \infty} 1/n^2$ is not in $(0,1]$, since it is equal to $0$.

The key point here is that the limit refers only to the final answer, not to the "process" of getting there. So, if you want to determine whether something is true of a limit, you only care about what number the limit is. The numbers you used to approach the limit are completely irrelevant.

The following analogy may be helpful. Suppose Alice has one brother, whose name is Bob. Suppose someone comes up to you and asks whether Alice's brother has brown hair. To determine this, you would want to look at Bob and see if he has brown hair. You wouldn't want to look at Alice's hair! In the same way, if someone asks you whether $\lim_{n\to \infty} 1/n^2$ is in $(0,1]$, you want to look at the value of the limit. You don't want to look at the value of the numbers $1/n^2$. The limit is defined in terms of those numbers, just as "Alice's brother" is defined in terms of "Alice", but the limit is just as distinct an entity from them as Bob is from Alice.

Answer 3

The interval $[0,1)$ is open at the right end therefore $$0\bar{.9} =1 \notin [0,1)$$

Similarly $$ \lim_{n\to \infty} 1/n^2 = 0 \notin (0,1],$$ because the interval $(0,1] $ does not include $0.$

That is a good way to define closed sets in general.

If the limit of every convergent sequence is in the set then the set is closed.

Thus none of $[0,1)$ or $(0,1]$ is closed.