Straight Line Segment is Shortest Path between Two Points/Lemma 1

Lemma to Straight Line Segment is Shortest Path between Two Points

Let $p, q \in \R$ be real numbers such that $\sqrt {1 + p^2} = \sqrt {1 + q^2}$.

Then:

$1 + \size p = 1 + \size q$

Proof

By definition of absolute value of a real number:

$\sqrt {p^2} = \size p$

Hence:

\(\ds \sqrt {p^2}\)

\(=\)

\(\ds \sqrt {q^2}\)

\(\ds \leadsto \ \ \)

\(\ds \size p\)

\(=\)

\(\ds \size q\)

\(\ds \leadsto \ \ \)

\(\ds 1 + \size p\)

\(=\)

\(\ds 1 + \size q\)

$\blacksquare$