## If $u_1,…,u_n$ are in $\mathbb R^n$ and the set $(u_i+e_i)$ is LD, show that $\sum_{i=0}^n ||u_i|| \ge 1$

The exercise is to show that $$\sum_{i=0}^n ||u_i|| \ge 1$$, knowing that the set of $$(u_i+e_i)$$ in $$\mathbb R^n$$ is LD, where $$e_i$$ is the canonical basis.

I know that $$e_i$$ are ortogonal vectors of modulus $$1$$, and that being LD means that at least one of the vectors $$(u_i+e_i)$$ can be expressed by a linear combination of the others. From this I tried isolating $$e_i$$, and thinking of inner product to use something like Cauchy-Schwarz to get an inequality, but was unable to develop the solution.