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 $\epsilon > 0$:

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

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

The result follows.