Real Addition Identity is Zero

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Theorem

The identity of real number addition is $0$:

$\exists 0 \in \R: \forall x \in \R: x + 0 = x = 0 + x$


Corollary

$\forall x, y \in \R: x + y = x \implies y = 0$


Proof

From the definition, the real numbers are the set of all equivalence classes $\eqclass {\sequence {x_n} } {}$ of Cauchy sequences of rational numbers.


Let $x = \eqclass {\sequence {x_n} } {}, y = \eqclass {\sequence {y_n} } {}$, where $\eqclass {\sequence {x_n} } {}$ and $\eqclass {\sequence {y_n} } {}$ are such equivalence classes.

From the definition of real addition, $x + y$ is defined as $\eqclass {\sequence {x_n} } {} + \eqclass {\sequence {y_n} } {} = \eqclass {\sequence {x_n + y_n} } {}$.

Let $\sequence {0_n}$ be such that $\forall i: 0_n = 0$.

Then we have:

\(\displaystyle \eqclass {\sequence {0_n} } {} + \eqclass {\sequence {x_n} } {}\) \(=\) \(\displaystyle \eqclass {\sequence {0_n + x_n} } {}\)
\(\displaystyle \) \(=\) \(\displaystyle \eqclass {\sequence {0 + x_n} } {}\)
\(\displaystyle \) \(=\) \(\displaystyle \eqclass {\sequence {x_n} } {}\)

Similarly for $\eqclass {\sequence {x_n} } {} + \eqclass {\sequence {0_n} } {}$.

Thus the identity element of $\struct {\R, +}$ is the real number $0$.

$\blacksquare$


Sources