It is relatively easy to prove that the set of perfect squares has asymptotic density equal to $0$. Then either the set $Q_2 := \{x^2+y^2: x,y \in \mathbf N\}$ has positive lower asymptotic density, and we're done, or the lower density of $Q_2$ is zero, and then we just consider that $\mathbf N = Q_2 + Q_2$ (by Lagrange's four-squares theorem).

By the way, it follows, e.g., from E. Landau, *Über die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindeszahl der zu ihrer additiven Zusammensetzung erforderlichen Quadrate*, Arch. Math. Phys. **13** (1908), 305-312 that the asymptotic density of $Q_2$ is actually zero, but this is more than what you need to answer the question in the OP.