Sequence Converges to Within Half Limit
Theorem
Sequence of Real Numbers
Let $\sequence {x_n}$ be a sequence in $\R$.
Let $\sequence {x_n}$ be convergent to the limit $l$.
That is, let $\ds \lim_{n \mathop \to \infty} x_n = l$.
Suppose $l > 0$.
Then:
- $\exists N: \forall n > N: x_n > \dfrac l 2$
Similarly, suppose $l < 0$.
Then:
- $\exists N: \forall n > N: x_n < \dfrac l 2$
Sequence of Complex Numbers
Let $\sequence {z_n}$ be a sequence in $\C$.
Let $\sequence {z_n}$ be convergent to the limit $l$.
That is, let $\ds \lim_{n \mathop \to \infty} z_n = l$ where $l \ne 0$.
Then:
- $\exists N: \forall n > N: \cmod {z_n} > \dfrac {\cmod l} 2$
Sequence in Normed Division Ring
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring with zero $0$.
Let $\sequence {x_n}$ be a sequence in $R$.
Let $\sequence {x_n}$ be convergent in the norm $\norm {\, \cdot \,}$ to the following limit:
- $\ds \lim_{n \mathop \to \infty} x_n = l \ne 0$
Then:
- $\exists N: \forall n > N: \norm {x_n} > \dfrac {\norm l} 2$
Also see
This is used in the Quotient Rule for Sequences.
Although this result seems a little trivial, it is often crucial to know that a sequence will be "eventually non-zero" so we know we can legitimately divide by it.