The Ring of Symmetric Functions over a commutative ring $R$, $\Lambda$, is the subring of the ring of formal power series $R[[x_1, x_2, \dots]]$ such that $f \in \Lambda$ if $f$ is invariant under every permutation of of the indeterminants and the degrees of the monomials occurring in $f$ are bounded.

Why do we need this second condition? I know the second condition gives that $\Lambda \cong R[\sigma_1,\sigma_2, \dots]$, the polynomial ring in the infinite indeterminants $\sigma_i$, but is there another reason we require this? Thanks!