How to prove that a sequence forms a Schauder basis for $\ell^\infty$?

Let $\alpha, \beta \in \mathbb{R}$ such that $0 < |\alpha / \beta| < 1$. Define the sequence of vectors in $\ell^\infty$ as follows:

$x_1 = ( \alpha , \beta , 0 , \ldots) , x_2 = (0,\alpha , \beta , 0 , \ldots ) , x_3 = (0,0,\alpha , \beta , 0 , \ldots )$,and so on.I want to show that the sequence $\{x_n\}_{n \in \mathbb{N}}$ forms a Schauder basis for $\ell^\infty$.

Using the representation $x = \displaystyle\sum_{n=1}^{\infty} a_n x_n$, I verified that the coordinates satisfy the recurrence:

$$x_n = a_n \alpha + a_{n+1} \beta$$which implies:

​$$ a_n = \dfrac{x_n - a_{n+1} \beta}{\alpha} $$My question is: How can I rigorously prove that for every $\varepsilon > 0$, there exists $N_0 \in \mathbb{N}$ such that for all $N \geq N_0$, we have$$ \left| \left| x- \displaystyle \sum_{n=1}^{N}a_n x_n \right| \right| < \epsilon ?$$I'm struggling to justify the convergence of the partial sums in the $\ell^\infty$ norm. Any help or insight would be appreciated!