Induction for a sequence starting with a negative and ending with a positive number.

Question

Prove by induction on n that for any $n \ge 2$, any sequence of non-zero real numbers $a_1, a_2, \dots, a_n$ that starts with a negative number (meaning $a_1 < 0$) and ends with a positive number (meaning $a_n > 0$) must have two consecutive elements $a_i$ and $a_{i+1}$ such that $a_i < 0$ and $a_{i+1} > 0$.

Answer 1

Well to begin at $n=2$, we have $a_1<0$ and $a_2>0$, so we're done.

Suppose that we know the $n-1$ case. Then consider a sequence of numbers $a_1,\ldots,a_n$. We have two situations. Suppose that $a_{n-1}>0$. Then we apply the inductive step to the series $a_1,\ldots,a_{n-1}$ to conclude. If $a_{n-1}<0$, then since we have $a_n>0$, so we're also done.

Answer 2

Hint. For the inductive step you have a sequence $a_1,\ldots,a_n$ where $a_1<0$ and $a_n>0$. Consider two cases:

• if $a_{n-1}<0$ there is an obvious case of what you want;
• if $a_{n-1}>0$ use the inductive hypothesis and think about the sequence $a_1,\ldots,a_{n-1}$.