\section*{Exercise 1 - 141221}

Let \(x_{n}(n=0,1,2,3, \ldots)\) be the sequence of numbers for which \(x_{0}=1\) and

\[
x_{n}=\frac{x_{n-1}}{x_{n-1}+1}
\]

\((n=1,2,3, \ldots)\) holds.

Determine the terms \(x_{1}, x_{2}\) and \(x_{3}\) of this sequence. Provide a term \(f(n)\) with the property \(f(n)=x_{n}(n=1,2,3, \ldots)\).
}

It holds that

\[
x_{0}=1, \quad x_{1}=\frac{1}{1+1}=\frac{1}{2}, \quad x_{2}=\frac{\frac{1}{2}}{\frac{1}{2}+1}=\frac{1}{3}, \quad x_{3}=\frac{\frac{1}{3}}{\frac{1}{3}+1}=\frac{1}{4}
\]

which suggests the conjecture that the general formula

\[
x_{n}=\frac{1}{n+1} \quad(n \in \mathbb{N})
\]

might hold. Indeed, it is clearly true for \(n=0\), and if it is already proven for some \(n \in \mathbb{N}\), then because

\[
x_{n+1}=\frac{x_{n}}{x_{n}+1}=\frac{\frac{1}{n+1}}{\frac{1}{n+1}+1}=\frac{1}{(n+1)+1}
\]

it also holds for \(n+1\), which thus proves its validity for all \(n \in \mathbb{N}\) by induction.