$$ S_{n}= x_{1} + x_{2} +...+ x_{n}, S^{*} _{n}=fl( S_{n} ) \\
S_{1} = x_{1}, S^{*} _{1}=fl( x_{1} )=x_{1} \\
S_{j+1} = S_{j} + x_{j+1}, S^{*} _{j+1}=fl(S^{*} _{j}+ x_{j+1} ) \\
p_{n+1} = \frac{S^{*} _{n+1}-S_{n+1}}{S_{n+1}} ,| \delta _{k} | \leq \varepsilon\\
p_{n+1} = \frac{fl(S^{*} _{n}+ x_{n+1})-(S_{n} + x_{n+1})}{S_{n+1}}\\
p_{n+1}= \frac{(S^{*} _{n}+ x_{n+1})(1+ \delta _{n} )-(S_{n} + x_{n+1})}{S_{n+1}} \\
p_{n+1}= \frac{(S_{n}(1+ \rho _{n} )+x_{n+1})(1+ \delta _{n} )-(S_{n} + x_{n+1})}{S_{n+1}}\\
p_{n+1}= \frac{S_{n} \rho _{n}+ \delta _{n}S_{n}+S_{n} \rho _{n} \delta _{n}+ x_{n+1} \delta _{n}}{S_{n+1}}\\
p_{n+1}= \frac{\delta _{n}(S_{n} + x_{n+1})+ \rho _{n}(S_{n}+S_{n}\delta _{n}}{S_{n+1}}\\
p_{n+1}= \frac{\delta _{n}S_{n+1}+ \rho _{n}S_{n}(1+\delta _{n})}{S_{n+1}} \\
p_{n+1}=\delta _{n}+ \rho _{n}( \frac{S_{n}}{S_{n+1}} )(1+\delta _{n})
$$
حالا چون اعداد ما مثبت بودن و $ S_{n} \leq S_{n+1} $ داریم
$$ |p_{n+1}| \leq \varepsilon +| \rho _{n}|(1)(1+ \varepsilon)= \varepsilon+ \Psi | \rho _{n}| s.t , \Psi =1+ \varepsilon $$
حالا داریم
$$ |p_{1}|=0 , |p_{2}| \leq \varepsilon , |p_{3}| \leq \varepsilon+ \Psi \varepsilon , |p_{4}| \leq \varepsilon+ \Psi \varepsilon+ \Psi ^{2} \varepsilon\\
|p_{n}| \leq \varepsilon+\Psi \varepsilon+...\Psi ^{n-1} \varepsilon= \varepsilon (1+ \Psi +...+ \Psi ^{n-1}) \\
|p_{n}| \leq \varepsilon ( \frac{ \Psi ^{n} -1}{ \Psi -1} )= \varepsilon ( \frac{ (1+ \varepsilon )^{n}-1 }{1+ \varepsilon -1} )= (1+ \varepsilon )^{n} -1 \simeq n \varepsilon
$$
