For the second one, merely check that the $\displaystyle k+1$st Newton iterate is $\displaystyle x_{k+1}=\frac{1}{2}\left(x_k+\frac{5}{x_k}\right)$. Now, I'm sure that you can prove that whatever your initial guess $\displaystyle x_0$ is, that this sequence will be monotone and bounded and so convergent.