I was doing excercises on convergence of sequences of real analysis but I came up with a problem I don't know how to prove.Note: The book has not shown Cauchy sequences yet and I don't know what they are.
Prove whether or not the sequence$$x_0=a:a>0$$$$x_{n+1}=x_n+\frac{1}{x_n}$$Is divergent or convergent.
I first wanted to prove that it is convergent by proving that it is monotonuously growing and that it is bounded.
I didn't know how to prove it is bounded, although I believe it's a divergent sequence. I first supposed that it is convergent, and since it is strictly increasing then I could use the fact that the limit of the sequence only depends on it's behavior on the long run to say that
$$lim(x_n)=lim(x_{n+1})$$ and$$x=x+\frac{1}{x} \implies 0 = \frac{1}{x}$$Which is only true if x approaches infinity.
I'm not sure if this works but I would like to know what is wrong with it rather than getting straight my question close right away please.
