Suranyi’s theorem

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

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

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

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

and we are done.