Is 1/3 included in the sequence 0.3, 0.33, 0.333,...?

Question

I assume that $\frac{1}{3}$ is equal to $0.3333...$.

Let's define a sequence as follows: $0.3$, $0.33$, $0.333$, $0.3333$,...

Question: is $\frac{1}{3}$ included in this sequence?

Every item in the sequence clearly has finite number of decimals, and $\frac{1}{3}$ has infinite decimals so it is clearly not included.

On the other hand the limit of this sequence is $\frac{1}{3}$ so it seems ok to say that it "includes" $\frac{1}{3}$.

What is the correct answer?

Answer 1

One thing that hasn't been said in the previous answers, and really needs to be emphasized, is that mathematics is not about guesswork, so we really cannot think "so it seems ok to ...". In mathematics, everything has to be precisely defined, and then whatever (precise) statement you make would be either true or false, and this is not up for debate.

Every item in the sequence clearly has finite number of decimals, and $1/3$ has infinite decimals so it is clearly not included.

Although not totally precise, you are essentially correct here. Precisely, each item in the sequence is equal to $k/10^m$ for some natural numbers $k,m$, but $1/3$ is not. We can prove this: Given any $k,m∈ℕ$ such that $1/3 = k/10^m$, we have $10^m = 3·k$, but $10^m = (9+1)^m$ $≡ 1^m = 1$ (mod $3$), whereas $3·k ≡ 0$ (mod $3$), so this is impossible.

On the other hand the limit of this sequence is $1/3$ so it seems ok to say that it "includes" $1/3$.

You can very well define "include" to mean something in order to make that sequence include its limit. But you must be very clear about the fact that it is not the conventional meaning of "include". So you were correct to use scare-quotes around "include", and it is important to realize how crucial precision is in mathematics.

Here is a standard mathematical notion that you may be interested in. Given any set $S ⊆ ℝ$, we say that $c$ is an adherent point of $S$ iff $c$ is a limit of some sequence from $S$. Given any sequence $f∈ℕ→ℝ$ and any $c∈ℝ$, it turns out that the following are equivalent:

• $c$ is an adherent point of $\{ \ f(k) : k∈ℕ \ \}$.
• There is some sequence $i∈ℕ→ℕ$ such that $f∘i$ converges to $c$.
• Either there is some increasing sequence $i∈ℕ→ℕ$ such that $f∘i$ converges to $c$ or there is some $k∈ℕ$ such that $f(k) = c$.

This notion may be the kind of relationship you are interested in, but it is not a good idea to call it "inclusion".

Answer 2

That is a series which converges to $\frac{1}{3}$ which, informally, means it will eventually get arbitrarily close. However, no entry in the series will be exactly $\frac{1}{3}$.

Similarly, the series $3$, $3.1$, $3.14$, $3,141$, $3.1415$ $3.14159$, ... converges to $\pi$ but no term is equal to $\pi$.

As Peter says in the comments, it would be better to say "sequence".

Answer 3

A single term of your series can be expressed as

$$A_n=\frac{1}{3}\bigg(1-\frac{1}{10^n}\bigg)$$

Using basic limits, this means that if $n→∞,A_n→\frac{1}{3}$. But according to your definition, $n$ is a countable number. Hence logically,$A_n<\frac{1}{3}$.

If you consider including the value even when n is $\infty$, then you may say it is included in the series, which isn't possible though. Like it depends upon your definition of "inclusion". Hope I was able to answer your question

Answer 4

As others have pointed out, this is a matter of definition. We might translate the statement that 1/3 is included in the sequence to something a little more precise, such as:

There exists an index such that the element of the sequence at that index is equal to 1/3.

In this case, if we're indexing by natural numbers, then the answer is NO (because it can be shown that every ℕ-indexed element of the sequence is distinct from 1/3).

However, a different choice would be to permit the index ω of a convergent sequence, in which case, the answer is YES, because it can be shown that the sequence converges to 1/3, and, by our definition, the element of the sequence at ω is that limit.

Now, this latter view is, I think, unconventional, but it is a choice that could be made.