Let $x > 0 \in \mathbb R$. Find sequences $\{a_n\}_{n=0}^{\infty}$, $\{b_n\}_{n=0}^{\infty}$ of natural numbers ($a_n, b_n \in \mathbb N$) such that $a_n + b_n = n$ and $a_n/b_n \rightarrow x$ for $n \rightarrow \infty$

I've proved using the squeeze theorem for sequences that $\frac 1 n\lfloor x n\rfloor \rightarrow x $ for $n \rightarrow \infty$. However I don't know how to apply this result to find the sequences in question.

Can anyone help me out ?

  • $\begingroup$ Sorry, once again. It should be $\frac 1 n \lfloor x n \rfloor \rightarrow x$ $\endgroup$ – Shuzheng Nov 21 '13 at 11:34

For each $n \geq 0$, pick $a_n$ and $b_n$ to satisfy $a_n + b_n = n$, $a_n/b_n = x$.

Substitution gives us \begin{align*} & b_n x + b_n = n \\ \implies & (x + 1) b_n = n \\ \implies & b_n = \frac{n}{x+1}. \end{align*}

And $a_n = n - b_n = n - \frac{n}{x+1} = \frac{nx}{x+1}$.

Edit: to get natural number sequences, I think we can take the floor of $a_n$ and the ceil of $b_n$, but we need to prove the limit of the ratio is unchanged.

Define sequences $q$ and $r$ by $\lfloor a_n \rfloor = a_n + q_n$, $\lceil b_n \rceil = b_n + r_n$.

Then \begin{align*} \frac{a_n +q_n}{b_n + r_n} &= \frac{a_n/b_n + q_n/b_n}{1 + r_n/b_n} \\ & \to x \qquad \text{as } n \to \infty. \end{align*} This proves that \begin{equation*} \frac{\lfloor a_n \rfloor}{\lceil b_n \rceil} \to x \end{equation*} as $n \to \infty$.

  • $\begingroup$ Hi little0 thanks for your answer. However I forgot to add that the sequences must consist of natural numbers only, sorry. $\endgroup$ – Shuzheng Nov 21 '13 at 11:32
  • $\begingroup$ Hmm, I bet we could take the floor of $a_n$ and the ceil of $b_n$ to get sequences of natural numbers with the desired properties. $\endgroup$ – littleO Nov 21 '13 at 11:40
  • $\begingroup$ I guess you are right. Right now I'm looking into it, but I've no experience calculating with floor and ceil in situations like this. $\endgroup$ – Shuzheng Nov 21 '13 at 11:52
  • $\begingroup$ I think we could prove a lemma that if $a_n \to \infty, b_n \to \infty$, and $\frac{a_n}{b_n} \to L$ as $n \to \infty$, then $\frac{a_n + q_n}{b_n + r_n} \to L$ as $n \to \infty$, assuming the sequences $q_n$ and $r_n$ are bounded. (Or if we like, we can assume $-1 \leq q_n \leq 1, -1 \leq r_n \leq 1$.) $\endgroup$ – littleO Nov 21 '13 at 11:54
  • $\begingroup$ @littleO : One may just take $\displaystyle b_n= \lfloor\frac{n}{x+1}\rfloor$ and $a_n= n-b_n$. By the result Nicolas Lykke Iversen stated we then have $\displaystyle a_n/b_n= \frac{n}{\lfloor\frac{n}{x+1}\rfloor}-1\to \frac{1}{1/(x+1)}-1=x$ as $n\to \infty$ $\endgroup$ – Teri Sep 7 '14 at 18:49

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