I've been puzzling with a math problem for a while and hope you can help me.
There is given a Hilbert space $H$ with an orthonormal basis $(v_i)_{i\in \mathbb{N}}$ and an increasing sequence $(\lambda_i)_{i > \in \mathbb{N}} \subset \mathbb{R}_0^+$, so $0<\lambda_1 \leq \lambda_2 \leq ... \rightarrow \infty$. Furthermore, we define the set $S:= \left\{ \sum_{i=1}^{\infty} \alpha_i v_i, \textit{where }\alpha_i > \in \mathbb{R} \textit{ and } \sum_{i=1}^{\infty} \alpha_i^2 \lambda_i\leq 1 \right\}$.
Show that $\inf_{E_n \subseteq H, dim(E_n)= n} \sup_{u \in S} \inf_{v \in E_n} ||u-v||_H = (\lambda_{n+1})^{-\frac{1}{2}}$.
I think I understood the statement itself and tried to show two inequalities, $\leq$ and $\geq$ to prove this fact. Unfortunately, neither of them was finished successfully.
I started with $||u-v||_H^2 = (u-v,u-v)_H$ and used the representation of $u\in S$ and $v \in E_n$ by the orthonormal basis. $u = \sum_{i=1}^{\infty} \alpha_i v_i$ with $\sum_{i=1}^{\infty} \alpha_i^2 \lambda_i \leq 1$ and $v = \sum_{k=1}^{n} \beta_k v_k$. Here I assumed, that basis vectors of $E_n$ are the first $n$ in $S$. But this assumption is just for convenience and can be removed if necesesary by a more complicated notation.
After simplifying the expression above by using the properties of the scalar product I got stuck since I didn't see a connection between the coefficients $\alpha$ and $\beta$ and although I can insert $\lambda_i$'s in the $(u,u)_X$-term I couldn't argue to get an expression with only $\lambda_{n+1}$ as required.
So maybe I am completely wrong or there is some (easy?) argument, that I don't see.
Does anyone have hints/ can help me with this problem?