I am not sure this belong to this site, in case I will post it elsewhere.
Let $P$ be the price of a bond, let $C_k$ the promised cash flow in year $k$. Then we define the interest yield $y$ as the real number ($y > -1$) such that
$$P = \sum_{k=1}^n \frac{C_k}{(1+y)^k}$$
In the definition is assumed that there exists only one $y$ that satisfies the above condition, but how to prove it?
Consider $$p(y) = \sum_{k=1}^n \frac{C_k}{(1+y)^k}$$ If we assume that all the $C_k$ are non-negative, and at least one $C_k > 0$, then:
Each term in the sum in $p(y)$ is a monotone decreasing function of $y$ for $y > -1$, so their sum $p(y)$ is a monotone strictly decreasing function of $y$.
$p(y)$ goes to infinity as $y$ approaches $-1$ from above.
Since $p(y)$ is a finite sum of terms each of which goes to zero as $y$ goes to infinity, $p(y)$ approaches zero as $y$ goes to infinity.
$p(y)$ is continuous.
Therefore, by the intermediate value theorem, for any positive finite value $P$ there is a value of $y$ such that $p(y) = P$. And this value, which we will call $Y(P)$, is unique because of for all $y > Y(P)$, $p(y) < P$ and for all $y < Y(P)$, $p(y) > P$ .
Now consider what happens if one or more of the $C_k < 0$. In that case, there can be multiple solutions for the effective rate of return.
For example, if the price is $1$ adn the returns from years 1 to 5 are $(3,5,-5,-8,5)$ the effective annual rate of return can be either $y=1.1472$ or $y=2.844$.
That is, this investment, with modest net long-term performance but a quick gin in the first couple of years, is fair either as a modest rate of return (if inflation is 14%) or in a hyper-inflation (184%) scenario in which those first two years of profit count for enough to balance the next two years of loss.