every basis is monotone w.r.t. an equivalent norm $\||{x} \|| = \sup_n \|S_n x\|$

Question

I want to prove that every basis $\{x_n\}$ is monotone with respect to the equivalent norm $\||x\|| = \sup_n \|S_n x\|$ where $S_n$ is the natural projections associated to the basis. All that i can move $$\||{S_N x}\|| = \displaystyle \sup_M \|S_M S_N x\| = \displaystyle \sup_{1 \leq M \leq N} \|S_M x\| \leq \||{x}\||.$$ So, $\displaystyle \sup_N \||{S_N} \|| \leq 1.$ How i can prove that $\|| S_N \|| \geq 1$ to prove that $\sup_n \||{S_N} \|| = 1.$

Answer 1

$||| S_Nx_1||| =|||x_1|||=\sup_n \|S_nx_1\|\geq \|S_1x_1\|=\|x_1\|=|||x_1|||$ and $x_1 \neq 0$. By definition of operator norm this implies $|||S_N||| \geq 1$.