How to show linear operator $T:\ell_1\rightarrow \ell_1$ such that $\lVert T\rVert =1$ never attains norm.

Question

I have a question concerning the following proof. Let linear operator $T:\ell_1\rightarrow\ell_1$ be defined as $$T(x_1,\ldots,x_i,\ldots)=((1-\frac{1}{1})x_1,\ldots,(1-\frac{1}{i})x_i,\ldots)$$ I have been able to show that $\lVert T\rVert=1$, but have difficulties showing that there is no $x\in \ell_1$ with $\lVert x\rVert_1=1$ such that $\lVert Tx\rVert_1=1$.

Answer 1

Hint:

If $0<a_i<b_i$ for all $i\in\mathbb N$, then

$$\sum_{i=1}^\infty a_i < \sum_{i=1}^\infty b_i$$

Hint 2:

Take any $x$, and write out the definition of $\|x\|_1$ and $\|Tx\|_1$.