Show surjectivity of Backward shifting $T: F^{\infty} \Rightarrow F^{\infty}$ defined as $T(x_1,x_2,...) =(x_2,x_3,....)$

Question

First what is the range of Backward shifting $T: F^{\infty} \rightarrow F^{\infty}$ defined as $T(x_1,x_2,...) =(x_2,x_3,....)$

Then to show that $T$ is surjective, I must show that $\operatorname{Range}(T)=F^\infty$.

And I know I have to show that $\operatorname{Range}(T) \subset F^\infty$ and $F^\infty \subset \operatorname{Range}(T)$. But I'm not able to show it.

So if someone can help be would be very helpful. Thanks in advance.

Answer 1

Hint: What is $T(0,x_1,x_2,x_3,\dots)$ ?