Existence of random variable given first $k$ moments
A less high-brow approach is to note that a $k\times k$ positive definite (strictly positive definite) Hankel matrix $A$ can be extended to a $(k+1)\times(k+1)$ one by appending a $(k+1)$st row and column, preserving positive definiteness. The Hankel structure determines all the entries in the extension except the entries $a_{k,k+1} = a_{k+1,k}$ and $a_{k+1,k+1},$ which I'll denote by $w$ and $z$, respectively. By Sylvester's criterion, it suffices to choose $(w,z)$ so the determinant $D(w,z)$ of the new matrix is positive. By expanding minors, say, we see $D$ is a linear function of $z$ plus an inhomogeneous quadratic function of $w$. Further, the coefficient of $z$ is the determinant of the original matrix, which by assumption is positive. So any choice of $w$ followed by a sufficiently large choice of $z$ makes $D(w,z)>0$.
(Thanks to @tristan for pointing out a flaw and its fix in an earlier version of this answer.)