**Edit:** My original idea doesn't work, but unknown's does. Here are the details.

Let $k$ be a field, which is WLOG algebraically closed. Let $V$ be a finite-dimensional representation over $k$ of dimension $n$. Then $S_{\infty}$ contains $(\mathbb{Z}/p\mathbb{Z}))^{p^n}$ (in fact any finite group) as a subgroup, where $p \neq \text{char}(k)$. Let $g_1, ... g_{p^n}$ be its generators. Then by "elementary linear algebra" $V$ is a direct sum of $1$-dimensional irreps of $\langle g_1 \rangle$ which $g_2, ... g_{p^n}$ must preserve, while still having order $p$. But there are only $p^n - 1$ nontrivial ways to do this; hence either one of the $g_i$ acts as the identity or two of them are the same and $V$ cannot be faithful.

Of course, whether this is "elementary linear algebra" or representation theory is debatable, and I think irrelevant. All I did was find a sequence of finite groups such that the dimension of the smallest faithful representation goes to infinity and I could have done this any number of ways, e.g. I could have chosen $\text{PSL}_2(\mathbb{F}_q)$.