Let $Lambda(x)=(lambda_1x_1,lambda_2x_n,...)$ be an operator $l_2 to l_2$. Show its range is closed iff $inf_{lambda_knot=0} |lambda_k|>0$

Hints: Let $M$ be the closed subspace generated by ${e_i: lambda_i neq 0}$ where $(e_i)$ is the standard orthonormal basis. (Note that $M$ consist of the sums $sum_{lambda_i neq 0} a_ie_i$ with $ sum |a_i|^{2} <infty$). Let $Lambda_1=Lambda |M$. Show that range of $Lambda_1$ is also closed and that $Lambda_1$ is also injective. By open mapping theorem its inverse is continuous so there exist a constant $C$ such that $sum_{lambda_i neq 0} |lambda_i x_i|^{2} geq C sum |x_i|^{2}$ for all $(x_i)$. From this the conclusion follows immediately.