Convergence by Multiple of Error Term

Theorem
Let $$\left \langle {s_n} \right \rangle$$ be a real sequence.

Suppose that $$\exists \epsilon \in \R, \epsilon > 0$$ such that:
 * $$\exists N \in \N: \forall n \ge N: \left|{s_n - l}\right| < K \epsilon$$

for any $$K \in \R, K > 0$$, independent of both $$\epsilon$$ and $$N$$.

Then $$\left \langle {s_n} \right \rangle$$ converges to $$l$$.

Proof
Let $$\epsilon > 0$$.

Then $$\frac \epsilon K > 0$$.

If the condition holds as stated, then:
 * $$\exists N \in \N: \forall n \ge N: \left|{s_n - l}\right| < K \left({\frac \epsilon K}\right)$$

Hence the result by definition of a convergent sequence.