How can i show the relationship between a sequence in a space in norm E with a genuinely increasing sequence of natural numbers?

Question

Supposing i have a sequence $(a_n)_{n\epsilon\mathbb{N}}$ in a space with norm E .

How can i show that a sequence $(x_n)_{n\epsilon\mathbb{N}}$ exists in E , so that $ \lim_{n\to\infty} x_n = 0 $ and a genuinely increasing sequence $(k_n)$ of natural numbers so that $a_n = x_0 + x_1 +x_2 +x_3+...+x_{k_n} \forall n\epsilon\mathbb{N}$

Answer 1

Hint

You need $$ x_{k_{n}+1}+...+ x_{k_{n+1}}=a_{n+1}-a_n $$ Try to pick inductively $k_{n+1}$ such that $$ \|\frac{a_{n+1}-a_n}{k_{n+1}-k_n} \| < \frac{1}{n} $$ and set $$ x_{k_{n}+1}=...= x_{k_{n+1}}=\frac{a_{n+1}-a_n}{k_{n+1}-k_n} $$