Cauchy Sequence with Finite Elements Prepended is Cauchy Sequence

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

Let $\sequence {x_n}$ be a sequence in $R$.

Let $N \in \N$

Let $\sequence {y_n}$ be the sequence defined by:
 * $\forall n, y_n = x_{N+n}$

Let $\sequence {y_n}$ be a Cauchy sequence in $R$.

Then:
 * $\sequence {x_n}$ is a Cauchy sequence in $R$.

Proof
Given $\epsilon \gt 0$

By the definition of a Cauchy sequence then:
 * $\exists N': \forall n, m \gt N', \norm {y_n - y_m} \lt \epsilon$

Hence $\forall n, m \gt \paren {N'+N}$:

The result follows.