Existence of an injection between $\mathbb{Z}$ and $[0,1]$ Is the statement that there is an injection between $\mathbb{Z}$ and $[0,1]$ true or false
$\mathbb{Z}\xrightarrow{(1-1)} [0,1]$
i've build it on the definition of cardinality that is if $\lvert A \lvert \leq \lvert B \lvert$
then there exists an injection $f:A\longrightarrow B$ and one knows that
$\lvert \mathbb{Z} \lvert \leq \lvert [0,1]\lvert$
thus there exists an injection
Consider the function $f : \Bbb Z \longrightarrow [0,1]$ defined by $$
f(n) = \left\{
        \begin{array}{ll}
            \dfrac {1} {1-2n} &amp; \quad n \leq 0 \\
            \dfrac {1} {2n} &amp; \quad n &gt; 0
        \end{array}
    \right.
$$
It is true, take for example a function $f:\mathbb{Z} \to [0,1]$ $$f(n)=\cases{{1\over 2|n|+1};\;\;\; n\leq0\\{1\over 2n};\;\;\; n&gt;0}$$