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 $\dfrac \epsilon K > 0$.

If the condition holds as stated, then:

$\exists N \in \N: \forall n \ge N: \left|{s_n - l}\right| < K \left({\dfrac \epsilon K}\right)$

Hence the result by definition of a convergent sequence.

$\blacksquare$

Sources

• W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Lemma $1.2.4$