The way the book proved it is this:
Its idea was to prove that the sequence is Cauchy then it follows that it is convergent in $\mathbb{R}$ ;
But my way that I want to check if it's right or wrong is :
$$ \begin{aligned} \left|U_{n+p}-U_{n}\right| &=\left|\frac{\cos 5^{n+1}}{5^{n+1}}+\frac{\cos 5^{n+2}}{5^{n+2}}+\cdots+\frac{\cos 5^{n+p}}{5^{n+p}}\right| \\ & \leq\left|\frac{\cos 5^{n+1}}{5^{n+1}}\right|+\left|\frac{\cos 5^{n+2}}{5^{n+2}}\right|+\cdots+\left|\frac{\cos 5^{n+p}}{5^{n+p}}\right| \\ & \leq\left|\frac{1}{5^{n+1}}\right|+\left|\frac{1}{5^{n+2}}\right|+\cdots+\left|\frac{1}{5^{n+p}}\right|=\frac{1}{5^{n+1}}+\frac{1}{5^{n+2}}+\cdots+\frac{1}{5^{n+p}} \\ &=\frac{1}{5^{n+1}} \frac{1-\left(\frac{1}{5}\right)^{p}}{1-\frac{1}{5}}=\frac{1}{5^{n+1}} \cdot \frac{5}{4} \cdot\left(1-\left(\frac{1}{5}\right)^{p}\right) \end{aligned} $$
Then I'll take the limit of $|U_{n+p}-U_{n}|$ as $n \rightarrow \infty$, which it's zero ; so isn't that enough to prove that the sequence is convergent in $\mathbb{R}$?
