Show that if $u$ is harmonic in $\mathbb{R}^n$ and $u = o(|x|)$, then $u$ is constant.

Question

Show that if $u$ is harmonic in $\mathbb{R}^n$ and $u = o(|x|)$, then $u$ is constant.

I am trying to prove this result by the mean value property. But it doesn't work. Please help me. Thanks!

PS: I just realized that one can prove it easily by the estimates on the derivatives of harmonic functions, which implies a stronger result:

If $u$ is harmonic in $\mathbb{R}^n$ and $u \le C(1+|x|)^N$, then $u$ is a polynomial of degree at most $N$.

But I am still curious about how to prove it directly by the mean value property, which is the "Hint" for this problem.