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In a joint paper that I am writing, we need (and prove) the following:

Lemma. Let $(a_n)_{n \ge 1}$ and $(b_n)_{n \ge 1}$ be sequences of positive real numbers such that:

   (i) $a_n + 1 \le b_n < a_{n+1}$ for all $n \in \mathbf{N}^+$;
   (ii) $a_n/b_n \to \ell$ as $n \to \infty$;
   (iii) $b_n/a_{n+1} \to 0$ and $n b_n/b_{n+1} \to 0$ as $n \to \infty$.

Then $\mathsf{d}^\ast(X) = 1-\ell$, where $X := \bigcup_{n \ge 1} [\![ a_n+1, b_n ]\!]$.

Here, for a set $X \subseteq \mathbf{Z}$ we let $\mathsf{d}^\ast(X)$ be the upper asymptotic density of $X$, viz. $$\mathsf{d}^\ast(X) := \limsup_{n \to \infty} \frac{|X \cap [\![ 1, n ]\!]|}{n},$$ and for $a,b \in \mathbf{R}$ we set $[\![a,b]\!] := [a,b] \cap \mathbf Z$.

The proof is rather straightforward, but at the same time it takes about half a page to word it in a reasonable amount of details. Yet, we would prefer to avoid it, so my question is:

Does any of you here around know of a reference where the lemma, or a generalization of it, is proved?

E.g., we have tried to look at Halberstam and Roth's Sequences (Springer-Verlag, 1989), but couldn't find anything.

Update. Based on Anthony Quas' comment below, the proof can be made sensibly shorter and the lemma can be slightly generalized by weakening the old assumption (iii).

In a joint paper that I am writing, we need (and prove) the following:

Lemma. Let $(a_n)_{n \ge 1}$ and $(b_n)_{n \ge 1}$ be sequences of positive real numbers such that:

   (i) $a_n + 1 \le b_n < a_{n+1}$ for all $n \in \mathbf{N}^+$;
   (ii) $a_n/b_n \to \ell$ as $n \to \infty$;
   (iii) $b_n/a_{n+1} \to 0$ as $n \to \infty$.

Then $\mathsf{d}^\ast(X) = 1-\ell$, where $X := \bigcup_{n \ge 1} [\![ a_n+1, b_n ]\!]$.

Here, for a set $X \subseteq \mathbf{Z}$ we let $\mathsf{d}^\ast(X)$ be the upper asymptotic density of $X$, viz. $$\mathsf{d}^\ast(X) := \limsup_{n \to \infty} \frac{|X \cap [\![ 1, n ]\!]|}{n},$$ and for $a,b \in \mathbf{R}$ we set $[\![a,b]\!] := [a,b] \cap \mathbf Z$.

The proof is rather straightforward, but at the same time it takes about half a page to word it in a reasonable amount of details (EDIT: this is no longer the case). Yet, we would prefer to avoid it, so my question is:

Do you know of a reference where the lemma, or a generalization of it, is proved?

E.g., we have tried to look at Halberstam and Roth's Sequences (Springer-Verlag, 1989), but couldn't find anything.

In a joint paper that I am writing, we need (and prove) the following:

Lemma. Let $(a_n)_{n \ge 1}$ and $(b_n)_{n \ge 1}$ be sequences of positive real numbers such that: (i) $a_n + 1 \le b_n < a_{n+1}$ for all $n \in \mathbf{N}^+$; (ii) $a_n/b_n \to \ell$, for some $\ell \in \mathbf{R}$, as $n \to \infty$;  (iii) $b_n/a_{n+1} \to 0$ and $n b_n/b_{n+1} \to 0$ as $n \to \infty$. Then $\mathsf{d}^\ast(X) = 1-\ell$, where $X := \bigcup_{n \ge 1} [\![ a_n+1, b_n ]\!]$.

Here, for a set $X \subseteq \mathbf{Z}$ we let $\mathsf{d}^\ast(X)$ be the upper asymptotic density of $X$, viz. $$\mathsf{d}^\ast(X) := \limsup_{n \to \infty} \frac{|X \cap [\![ 1, n ]\!]|}{n},$$ and for $a,b \in \mathbf{R}$ we set $[\![a,b]\!] := [a,b] \cap \mathbf Z$.

The proof is, to some extent, rather straightforward, but at the same time it takes about half a page to word it in a reasonable amount of details. Yet, we would prefer to avoid it, so my question is:

Does any of you here around know of a reference where the lemma, or a generalization of it, is proved?

E.g., we have tried to look at Halberstam and Roth's Sequences (Springer-Verlag, 1989), but couldn't find anything.

In a joint paper that I am writing, we need (and prove) the following:

Lemma. Let $(a_n)_{n \ge 1}$ and $(b_n)_{n \ge 1}$ be sequences of positive real numbers such that: 
 
   (i) $a_n + 1 \le b_n < a_{n+1}$ for all $n \in \mathbf{N}^+$;  
   (ii) $a_n/b_n \to \ell$ as $n \to \infty$; 
   (iii) $b_n/a_{n+1} \to 0$ and $n b_n/b_{n+1} \to 0$ as $n \to \infty$. 

Then $\mathsf{d}^\ast(X) = 1-\ell$, where $X := \bigcup_{n \ge 1} [\![ a_n+1, b_n ]\!]$. 

Here, for a set $X \subseteq \mathbf{Z}$ we let $\mathsf{d}^\ast(X)$ be the upper asymptotic density of $X$, viz. $$\mathsf{d}^\ast(X) := \limsup_{n \to \infty} \frac{|X \cap [\![ 1, n ]\!]|}{n},$$ and for $a,b \in \mathbf{R}$ we set $[\![a,b]\!] := [a,b] \cap \mathbf Z$.

The proof is rather straightforward, but at the same time it takes about half a page to word it in a reasonable amount of details. Yet, we would prefer to avoid it, so my question is:

Does any of you here around know of a reference where the lemma, or a generalization of it, is proved?

E.g., we have tried to look at Halberstam and Roth's Sequences (Springer-Verlag, 1989), but couldn't find anything.

In a joint paper that I'm writing, we need (and prove) the following:

Lemma. Let $(a_n)_{n \ge 1}$ and $(b_n)_{n \ge 1}$ be sequences of positive real numbers such that: (i) $a_n + 1 \le b_n < a_{n+1}$ for all $n \in \mathbf{N}^+$; (ii) $a_n/b_n \to \ell$, for some $\ell \in \mathbf{R}$, as $n \to \infty$; (iii) $b_n/a_{n+1} \to 0$ and $n b_n/b_{n+1} \to 0$ as $n \to \infty$. Then $\mathsf{d}^\ast(X) = \ell + 1$, where $X := \bigcup_{n \ge 1} [\![ a_n+1, b_n ]\!]$.

Here, for a set $X \subseteq \mathbf{Z}$ we let $\mathsf{d}^\ast(X)$ be the upper asymptotic density of $X$, viz. $$\mathsf{d}^\ast(X) := \limsup_{n \to \infty} \frac{|X \cap [\![ 1, n ]\!]|}{n},$$ and for $a,b \in \mathbf{R}$ we set $[\![a,b]\!] := [a,b] \cap \mathbf Z$.

The proof is, to some extent, rather straightforward, but at the same time it takes about half a page to word it in a reasonable amount of details). Yet, I would prefer to avoid it, so my question is:

Does any of you here around know of a reference where the lemma, or a generalization of it, is proved?

E.g., I've tried to look at Halberstam and Roth's Sequences (Springer-Verlag, 1989), but couldn't find anything.

In a joint paper that I am writing, we need (and prove) the following:

Lemma. Let $(a_n)_{n \ge 1}$ and $(b_n)_{n \ge 1}$ be sequences of positive real numbers such that: (i) $a_n + 1 \le b_n < a_{n+1}$ for all $n \in \mathbf{N}^+$; (ii) $a_n/b_n \to \ell$, for some $\ell \in \mathbf{R}$, as $n \to \infty$; (iii) $b_n/a_{n+1} \to 0$ and $n b_n/b_{n+1} \to 0$ as $n \to \infty$. Then $\mathsf{d}^\ast(X) = 1-\ell$, where $X := \bigcup_{n \ge 1} [\![ a_n+1, b_n ]\!]$.

Here, for a set $X \subseteq \mathbf{Z}$ we let $\mathsf{d}^\ast(X)$ be the upper asymptotic density of $X$, viz. $$\mathsf{d}^\ast(X) := \limsup_{n \to \infty} \frac{|X \cap [\![ 1, n ]\!]|}{n},$$ and for $a,b \in \mathbf{R}$ we set $[\![a,b]\!] := [a,b] \cap \mathbf Z$.

The proof is, to some extent, rather straightforward, but at the same time it takes about half a page to word it in a reasonable amount of details. Yet, we would prefer to avoid it, so my question is:

Does any of you here around know of a reference where the lemma, or a generalization of it, is proved?

E.g., we have tried to look at Halberstam and Roth's Sequences (Springer-Verlag, 1989), but couldn't find anything.