Translation of Integer Interval is Bijection

Theorem
Let $a,b,c\in\Z$ be integers.

Let $\left[{a \,.\,.\, b}\right]$ denote the integer interval between $a$ and $b$.

Then the mapping $T : \left[{a \,.\,.\, b}\right] \to \left[{a+c \,.\,.\, b+c}\right]$ defined as:
 * $T(k) = T(k+c)$

is a bijection.

Proof
Note that if $k \in \left[{a \,.\,.\, b}\right]$, then indeed $k+c \in \left[{a+c \,.\,.\, b+c}\right]$.

Injectivity
Let $k,l \in \left[{a \,.\,.\, b}\right]$ with $k + c = l + c$.

By Integer Addition is Cancellable, $k = l$.

Thus $T$ is injective.

Surjectivity
Let $m \in \left[{a+c \,.\,.\, b+c}\right]$.

Then $m-c \in \left[{a \,.\,.\, b}\right]$.

Then $T(m-c) = m-c+c = m$.

Thus $T$ is surjective.

Also see

 * Cardinality of Integer Interval