Closedness of the completely positive cone

Use the conic Carathéodory theorem in the real space of symmetric matrices, of dimension $N=n(n+1)/2$. A convergent sequence $B_{\ell }$ in the completely positive cone has padded factorizations $B_{\ell }={\sum }_{j=1}^N{x}_{\ell j}{x}_{\ell j}^T$ with nonnegative factors. Their total squared norms equal $\mathrm{tr}{B}_{\ell }$ and are uniformly bounded. A simultaneous convergent subsequence of the finite factor tuple gives $B={\sum }_j{x}_j{x}_j^T$ with $x_j\ge 0$. Thus the limit remains in the cone. The argument also bounds the required number of factors by $N$.