prove that f is injective

Let

$ bv_0 = \{(x_k)\in c_0 |\sum_{i=1}^\infty |x_{k+1} -x_k| < \infty \}$

$l_1 := \{ (x_k)\in c_0 |\sum_{i=1}^\infty |x_k| < \infty \} $

We define

$ f: l_1 \to bv_0 $

$ f((x_k))= (x_{k+1} -x_k )$

I tried to prove that f is Injective function. I tried with definition of Injective function like this

let $ f((x_k))=f((y_k)) \Rightarrow (x_{k+1} -x_k ) = (y_{k+1} -y_k ) \Rightarrow \forall k \in N, x_{k+1} -x_k=y_{k+1} -y_k $

But I don't Know how I can continue!Also, I tried to prove like this :

Let $ (x_k) \neq (y_k) \Rightarrow \exists k \in N : x_k \neq y_k $

But again I don't Know how I can continue!

any help