Prove that every symmetric polynomial can be written in terms of the elementary symmetric polynomials

How do I prove that

any symmetric polynomial P is given by an expression involving only additions and multiplication of constants and elementary symmetric polynomials.

I have no clue of where to start, I just know the basic definition:

The polynomial $P(x_1,x_2,...,x_n)$ is symmetric if for any permutation $\sigma$ of $\{x_1,x_2,...,x_n\}$, $$P(x_1,x_2,...,x_n)=P(x_{\sigma_1},x_{\sigma_2},...,x_{\sigma_n})$$ The elementary symmetric polynomials for a polynomial consists of $n$ variables, $\{x_1,x_2,...,x_n\}$, and are defined as $\{e_1,e_2,...,e_n\}$: $$e_0=1\\ e_1=\sum_{i}x_i\\ e_2=\sum_{i,j}x_ix_j\\ e_3=\sum_{i,j,k}x_ix_jx_k\\ ....................\\ e_n=x_1x_2...x_n$$

Can I use induction, if possible where do I start ?