Proofs:

One uses the reparametrization theorem to prove (ii) and (iii). Proof of (i) is more involved and we indicate two different methods by which this can be achieved. On the boundary $\dot{I}^2$ of the unit square $I^2$ we define a map $\phi: \dot{I}^2 \longrightarrow [0, 1]$ as follows.

$$\phi(0, t) = 0, \quad \phi(s, 0) = 0,\quad \phi(s, 1) = 0$$

Along the part $(1, t)$ of the boundary,

$$\phi(1, t) = \left\{\begin{array}{lll} 2t & & 0 \leq t \leq 1/2 \\ 2 - 2t & & 1/2 \leq t \leq 1 \\ \end{array} \right.$$

By Tietze's extension theorem $\phi$ extends continuously to $I^2$ taking values in $[0, 1]$. Consider now the map $H: I^2\longrightarrow X$ given by

$$H(s, t) = \gamma\circ\phi(s, t).$$

It is readily checked that $H$ establishes a homotopy between $\gamma*\gamma^{-1}$ and the constant path $\varepsilon_{\gamma(0)}$. $\square$