Please help me out reviewing the way I wrote this proof:
Prove by induction: $1^3+2^3+3^3+...+n^3=\left(\frac{n(n+1)}{2}\right)^2$ with $n\geqslant1$
Proof:
Lets define the set, $S=\left \{n\in N:n\geqslant1, 1^3+2^3+3^3+...+n^3=\left(\frac{n(n+1)}{2}\right)^2 \right \}$
If $n=1$ then, $1\in S$
Lets asume that $k\in S$ with $k\geqslant1$, then
$\begin{gather*} 1^3+2^3+3^3+...+k^3=\left(\frac{k(k+1)}{2}\right)^2\\ \end{gather*}$
Now lets proof that $k+1\in S$,
$\begin{align*}1^3+2^3+3^3+...+k^3+(k+1)^3=&\left(\frac{k(k+1)}{2}\right)^2+(k+1)^3\\ =&(k+1)^2\left (\frac{k^2}{4}+(k+1)\right)\\=&(k+1)^2 \left(\frac{(k^2+4k+4)}{4} \right)\\ =&(k+1)^2 \left(\frac{(k+2)^2}{4}\right)\\ =&\left(\frac{(k+1)(k+2)}{2}\right)^2 \end{align*}$
Which is true.
