Prove the positive definiteness of Hilbert matrix

Let $X=(x_i)_{1\leq i\leq n} \in \cal{M}_{n,1}(\mathbb{R}).$ We have $$ {}^tXAX=\sum_{1\leq i,j\leq n}\frac{x_ix_j}{i+j-1}=\sum_{1\leq i,j\leq n}x_ix_j\int_0^1t^{i+j-2}dt=\int_0^1\left(\sum_{i=1}^nx_it^{i-1}\right)^2dt>0 $$ for $X\neq0$, giving the announced result since $A$ is symmetric.

Let $H_n$ be the n-th order Hilbert matrix. To prove $H_n$ is positive defined, it suffices to show all the principal minor determinant of $H_n$ are positive. Say, $\det(H_m)>0$ for all $0\leq m\leq n$. This is true by the properties of Hilbert matrix. (see Hilbert Matrix).