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Set $U_1 = \{1,2\}$, and for $n\in\mathbb{N}$ with $n\geq 1$ we set $U_{n+1} = U_n\cup A_n$ where $A_n$ is the set of elements of $\mathbb{N}$ that are not in $U_n$ but can be uniquely written as a sum of $2$ elements of $U_n$.

Set $U=\bigcup_{n\in\mathbb{N}} U_n$.

What is the value of $$\lim \inf_{n\to\infty}\frac{|U\cap\{1,\ldots,n\}|}{n}$$?

EDIT: The sequence $\frac{|U\cap \{1,\ldots, n\}|}{n}$ seems to have a trend to grow smaller as $n$ grows large; I conjecture that the value I look for might be $0$.

Set $U_1 = \{1,2\}$, and for $n\in\mathbb{N}$ with $n\geq 1$ we set $U_{n+1} = U_n\cup A_n$ where $A_n$ is the set of elements of $\mathbb{N}$ that are not in $U_n$ but can be uniquely written as a sum of $2$ elements of $U_n$.

Set $U=\bigcup_{n\in\mathbb{N}} U_n$.

What is the value of $$\lim \inf_{n\to\infty}\frac{|U\cap\{1,\ldots,n\}|}{n}$$?

EDIT: The sequence $\frac{|U\cap \{1,\ldots, n\}|}{n}$ seems to have a trend to grow smaller as $n$ grows large; I conjecture that the value I look for might be zero.

Set $U_1 = \{1,2\}$, and for $n\in\mathbb{N}$ with $n\geq 1$ we set $U_{n+1} = U_n\cup A_n$ where $A_n$ is the set of elements of $\mathbb{N}$ that are not in $U_n$ but can be uniquely written as a sum of $2$ elements of $U_n$.

Set $U=\bigcup_{n\in\mathbb{N}} U_n$.

What is the value of $$\lim \inf_{n\to\infty}\frac{|U\cap\{1,\ldots,n\}|}{n}$$?

EDIT: The sequence $\frac{|U\cap \{1,\ldots, n\}|}{n}$ seems to have a trend to grow smaller as $n$ grows large; I conjecture that the value I look for might be $0$. But in the end of the day I have no idea.

Set $U_1 = \{1,2\}$, and for $n\in\mathbb{N}$ with $n\geq 1$ we set $U_{n+1} = U_n\cup A_n$ where $A_n$ is the set of elements of $\mathbb{N}$ that are not in $U_n$ but can be uniquely written as a sum of $2$ elements of $U_n$.

Set $U=\bigcup_{n\in\mathbb{N}} U_n$.

What is the value of $$\lim \inf_{n\to\infty}\frac{|U\cap\{1,\ldots,n\}|}{n}$$?

EDIT: The sequence $\frac{|U\cap \{1,\ldots, n\}|}{n}$ seems to have a trend to grow smaller as $n$ grows large; I conjecture that the value I look for might be $0$.

Set $U_1 = \{1,2\}$, and for $n\in\mathbb{N}$ with $n\geq 1$ we set $U_{n+1} = U_n\cup A_n$ where $A_n$ is the set of elements of $\mathbb{N}$ that are not in $U_n$ but can be uniquely written as a sum of $2$ elements of $U_n$.

Set $U=\bigcup_{n\in\mathbb{N}} U_n$.

What is the value of $$\lim \inf_{n\to\infty}\frac{|U\cap\{1,\ldots,n\}|}{n}$$?

Set $U_1 = \{1,2\}$, and for $n\in\mathbb{N}$ with $n\geq 1$ we set $U_{n+1} = U_n\cup A_n$ where $A_n$ is the set of elements of $\mathbb{N}$ that are not in $U_n$ but can be uniquely written as a sum of $2$ elements of $U_n$.

Set $U=\bigcup_{n\in\mathbb{N}} U_n$.

What is the value of $$\lim \inf_{n\to\infty}\frac{|U\cap\{1,\ldots,n\}|}{n}$$?

EDIT: The sequence $\frac{|U\cap \{1,\ldots, n\}|}{n}$ seems to have a trend to grow smaller as $n$ grows large; I conjecture that the value I look for might be $0$. But in the end of the day I have no idea.