Exercise 7.1.5 ($\langle a_0,a_1,\ldots,a_n \rangle \langle a_0,a_1,\ldots,a_n + c \rangle$ if $n$ odd.)

Question

Let $a_1,a_2,\ldots,a_n$ and $c$ be positive real numbers. Prove that
$$\langle a_0,a_1,\ldots,a_n \rangle > \langle a_0,a_1,\ldots,a_n + c \rangle$$
holds if $n$ is odd, but is false if $n$ is even.

Richard Ganaye · Accepted Answer

ewcommand{\Z}{\mathbb{Z}} ewcommand{\N}{\mathbb{N}} ewcommand{\R}{\mathbb{R}} \begin{proof} We prove this property by induction. If $n = 0$, then $\langle a_0 angle = a_0 < a_0 + c =\langle a_0 + c angle$. If $n = 1$, then $\langle a_0 ,a_1 angle > \langle a_0, a_1 + c angle$ by Problem 4. We define $\mathscr{P}(k)$ by \begin{align*} \mathscr{P}(k) \iff&\forall c \in \R^*,\ \forall (a_0,a_1,\ldots,a_{2k+1}) \in (\R^*)^{2k+2},\ &\left\{ \begin{array}{ll} \langle a_0, a_1,\ldots a_{2k} angle &<\langle a_0,a_1,\ldots,a_{2k} + c angle,\ \langle a_0,a_1,\ldots,a_{2k+1} angle &> \langle a_0,a_1,\ldots,a_{2k}+ c angle. \end{array} ight. \end{align*} We have verified that $\mathscr{P}(0)$ is true. Suppose now that $\mathscr{P}(k)$ is true, and let $a_0,a_1,\ldots,a_{2k+2}, a_{2k+3}$ be positive real numbers. Applying $\mathscr{P}(k)$ to the $(2k+2)$-uple $(a_1,a_2,\ldots, a_{2k+2})$, we know that for any $c>0$, $$\langle a_1,a_2,\ldots,a_{2k+2} angle > \langle a_1,a_2,\ldots,a_{2k+2}+ c angle>0.$$ Therefore \begin{align*} \langle a_0, a_1,\ldots, a_{2k+2} angle &= a_0 + \frac{1}{\langle a_1,\ldots, a_{2k+2} angle }\ &a_0 + \frac{1}{\langle a_1,\ldots, a_{2k+3}+c angle }\ &=\langle a_0, a_1,\ldots, a_{2k+3} angle, \end{align*} so \begin{align} \langle a_0, a_1,\ldots, a_{2k+3} angle > \langle a_0, a_1,\ldots, a_{2k+3} angle. \end{align} By (1) and (2), we obtain that $\mathscr{P}(k+1)$ is true, and the induction is done. This shows that if $a_0,a_1,\ldots,a_n$ and $c$ are positive real numbers, then \begin{align*} \langle a_0,a_1,\ldots,a_n angle &> \langle a_0,a_1,\ldots,a_n + c angle ext{ if $n$ is odd,}\ \langle a_0,a_1,\ldots,a_n angle &< \langle a_0,a_1,\ldots,a_n + c angle ext{ if $n$ is even.} \end{align*} Unfortunately, the statement don't assume that $a_0>0$, so we apply again the same argument: if $a$ is any real number, and $a_1,\ldots,a_n$ positive real numbers, since \begin{align*} \langle a_1,\ldots,a_n angle &> \langle a_1,\ldots,a_n + c angle ext{ if $n-1$ is odd,}\ \langle a_1,\ldots,a_n angle &< \langle a_1,\ldots,a_n + c angle ext{ if $n-1$ is even.} \end{align*} then \begin{align*} a_0 + \frac{1}{\langle a_1,\ldots,a_n angle} &< a_0 + \frac{1}{\langle a_1,\ldots,a_n + c angle} ext{ if $n-1$ is odd,}\ a_0 + \frac{1}{\langle a_1,\ldots,a_n angle} &>a_0 + \frac{1}{\langle a_0,a_1,\ldots,a_n + c angle} ext{ if $n-1$ is even,} \end{align*} so \begin{align*} \langle a_0,a_1,\ldots,a_n angle &< \langle a_0,a_1,\ldots,a_n + c angle ext{ if $n$ is even},\ \langle a_0,a_1,\ldots,a_n angle &> \langle a_0,a_1,\ldots,a_n + c angle ext{ if $n$ is odd.} \end{align*} (even if $a_0 <=0$). \end{proof}

Exercise 7.1.5 ($\langle a_0,a_1,\ldots,a_n \rangle > \langle a_0,a_1,\ldots,a_n + c \rangle$ if $n$ odd.)

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Exercise 7.1.5 ($\langle a_0,a_1,\ldots,a_n \rangle > \langle a_0,a_1,\ldots,a_n + c \rangle$ if $n$ odd.)

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