Let $X_i \in L_2$ be a sequence of pairwise correlated random variables. The random variables can have arbitrary positive correlation but can't have arbitrary negative correlation.
How can I show that for $(X_1,\ldots,X_n)$ and $\forall i,j \in \{1,\ldots,n\}, i\neq j$
$$\mathrm{Cor}(X_i,X_j)<\frac{-1}{n-1}$$ is not possible
The idea is to realize that covariance matrices are positive semi-definite. Define variables $Y_j = X_j/ \sqrt{\operatorname{Var}(X_j)}.$ Seeking a contradiction, suppose that the condition you mention holds, i.e. $$\operatorname{Cor}(X_i,X_j) = \operatorname{Cov}(Y_i,Y_j) < -\frac{1}{n-1}.$$
Let $\Sigma$ be the covariance matrix of $(Y_1,\ldots,Y_n)$. Then $\Sigma$ has $1$'s on the diagonal, and each off-diagonal entry is less than $-\frac{1}{n-1}$. If we define $x = (1,1,\ldots,1)^T$, then we see that $x^T \Sigma x < 0$, which contradicts the fact that covariance matrices are positive semi-definite.