Leigh.Samphier/Sandbox/Representative of P-adic Sum

Theorem

Let $p$ be any prime number.

Let $\Q_p$ be the $p$-adic numbers as a quotient of Cauchy sequences.

That is, $\Q_p$ is the quotient ring $\CC \, \big / \NN$ where:

$\CC$ denotes the commutative ring of Cauchy sequences over $\struct {\Q, \norm {\,\cdot\,}_p}$

$\NN$ denotes the set of null sequences in $\struct {\Q, \norm {\,\cdot\,}_p}$.

and $\norm {\,\cdot\,}_p$ denotes the p-adic norm on the rationals $\Q$.

Let $x, y$ be any left cosets of $\Q_p$.

Let $\sequence{x_n}$ and $\sequence{y_n}$ be any repesentatives of $x$ and $y$ respectively.

Then:

the sequence $\sequence{x_n + y_n}$ is a repesentative of $x + y$

Proof

$\blacksquare$