Let $B \in \mathbb{R}^{n \times n}$ be symmetric and positive semi-definite such that $B = U\Lambda U^T$, where $U = [u_1,\cdots,u_n]^T$ is an orthogonal matrix with $u_i \in \mathbb{R}^n$, and $\Lambda=\text{diag}(0,\cdots,0,\lambda_k,\cdots,\lambda_n)$ is the diagonal matrix with $0<\lambda_k \leq\cdots \leq \lambda_n$. Show that for $\Delta >0$ the following are equivalet:
1- $u_i^Tc=0,\,\,\,\,\,\, \forall i=1,\cdots,k-1$, and $\sum_{i=k}^n \frac{(u_i^T)^2}{\lambda_i^2}\leq \delta^2$
2- There exist $x \in \mathbb{R}^n$ such that $Bx+c = 0$ and and $\|x\| \leq \Delta$.
I can proof $(2)$ implies $(1)$ but do not know how to show $(1)$ implies $(2)$.