The sequence can be expressed as...

$\displaystyle \Delta_{n} = a_{n+1} - a_{n} = f(a_{n}) = 1 + \sqrt{a_{n}} - a_{n}$ (1)

The function that generates the sequence...

$\displaystyle f(x) = 1 + \sqrt{x} - x$ (1)

... is represented in figure...

... and, because is has only one fixed point at $\displaystyle x_{0} = \frac{3 + \sqrt{5}}{2} = 2,6180339887\dots$ and that is an attractive fixed point, any $\displaystyle a_{0} \ge 0$ will produce a sequence convergent at $\displaystyle x_{0}$ without oscillations, because the slope of $\displaystyle f(x)$ in $\displaystyle x=x_{0}$ is in absolute value less than $\displaystyle 1$...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$