Null Sequence Test for Convergence

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Theorem

Let $\struct {R, \norm{\,\cdot\,}}$ be a normed division ring.

Let $\sequence{x_n}$ be a convergent sequence in $\struct {R, \norm{\,\cdot\,}}$ with limit $l$.

Let $\sequence{y_n}$ be a sequence.

Then:

$\sequence{y_n}$ converges to the limit $l$ if and only if the sequence $\sequence{y_n - x_n}$ is a null sequence

Proof

Necessary Condition

Let $\sequence{y_n}$ converge to the limit $l$.

From Difference Rule for Sequences in Normed Division Ring:

$\ds \lim_{n \mathop \to \infty} y_n - x_n = l - l = 0$

Hence $\sequence{y_n - x_n}$ is a null sequence by definition.

$\Box$

Sufficient Condition

Let $\sequence{y_n - x_n}$ be a null sequence.

By definition of a null sequence:

$\ds \lim_{n \mathop \to \infty} y_n - x_n = 0$

From Sum Rule for Sequences in Normed Division Ring:

$\ds \lim_{n \mathop \to \infty} x_n + \paren{y_n - x_n} = l + 0 = l$

For all $n \in \N$:

$x_n + \paren{y_n - x_n} = y_n$

Hence:

$\ds \lim_{n \mathop \to \infty} y_n = l$

It follows that $\sequence{y_n}$ converges to the limit $l$ by definition.

$\blacksquare$