Definition: Let $X, Y$ be Banach spaces. Then we call a linear map $T: X \to Y$ a compact operator if $\overline{T(K_1^X(0))}$ is compact in $Y$.
$K_1^X(0)$ denotes the ball with radius $1$ and center $0$ in $X$.
Proposition: Every $T \in \mathcal B(X,Y)$ with $\mathrm{dim\ ran} \ T < \infty$ is compact.
Proposition: The set $\mathcal K(X,Y)$ of all compact operators in $\mathcal B(X,Y)$ is a closed linear subspace of $\mathcal B(X,Y)$ with respect to the operator norm.
Task:
Consider the decreasing sequence $(a_n)_{n \in \mathbb N} \in l^1(\mathbb N)$ and the map $A:l^2(\mathbb N) \to l^2(\mathbb N)$ defined as $(x_n)_{n \in \mathbb N}\mapsto (\sum_{k=1}^{\infty}a_{k+n-1}x_k)_{n \in \mathbb N}$
I could show that $\|Ax\|_{l^2(\mathbb N)}^2\leq\|x\|_{l^2(\mathbb N)}^2\|a\|_{l^1(\mathbb N)}^2<\infty$.
So, $A$ is well defined and bounded. Also it is clearly linear. I want to use the first proposition.
How can I show that $\mathrm{dim\ ran} \ A < \infty$? I don't really know how/where to start.
