Suranyi’s theorem

Suranyi’s theorem states that every integer $k$ can be expressed as the following sum:

$$k=\pm 1^2\pm 2^2\pm \mathrm{\cdots }\pm m^2$$

for some $m\in {\mathbb{Z}}^{+}$.
We prove this by induction, taking the first four whole numbers as our cases:

$$0=1^2+2^2-3^2+4^2-5^2-6^2+7^2$$

$$1=1^2$$

$$2=-1^2-2^2-3^2+4^2$$

$$3=-1^2+2^2$$

Now it suffices to prove that if the theorem is true for $k$ thenit is also true for $k+4$.
As

$${(m+1)}^2-{(m+2)}^2-{(m+3)}^2+{(m+4)}^2=4$$

it’s simple to finish the proof:
if $k=\pm 1^2\pm \mathrm{\cdots }\pm m^2$ then

$$(k+4)=\pm 1^2\pm \mathrm{\cdots }\pm m^2+{(m+1)}^2-{(m+2)}^2-{(m+3)}^2+{(m+4)}^2$$

and we are done.