Can anyone give me a satisfactory proof that the real sequence $(x_n)$ defined by $x_n = 2^n - n$ diverges to $+\infty$?
The heuristic reason is that $ \lim_{n\to\infty} \frac{n}{2^n} = 0, $ but I can't seem to turn this into a rigorous proof.
More generally is there a theorem which says that $(z_n-y_n)$ diverges to $+\infty$ if $(y_n)$ and $(z_n)$ both diverge to $+\infty$ and $\lim_{n\to\infty} y_n/z_n = 0$?