# Convergence by Multiple of Error Term

## Theorem

Let $\sequence {s_n}$ be a real sequence.

Suppose that $\exists \epsilon \in \R, \epsilon > 0$ such that:

$\exists N \in \N: \forall n \ge N: \size {s_n - l} < K \epsilon$

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

Then $\sequence {s_n}$ converges to $l$.

## Proof

Let $\epsilon > 0$.

Then $\dfrac \epsilon K > 0$.

If the condition holds as stated, then:

$\exists N \in \N: \forall n \ge N: \size {s_n - l} < K \paren {\dfrac \epsilon K}$

Hence the result by definition of a convergent sequence.

$\blacksquare$