I aim to prove that homotopy is a transitive property. If anybody could let me know if the following proof that I have constructed is correct, I would greatly appreciate it.
Let $\delta(t), \eta(t), \sigma(t)$ be three continuous well defined paths on an open set $U$ s.t. they are all defined for $t\in[a,b]$.
Suppose that $\alpha(t,h):[a,b]\times[c_1,d_1]\rightarrow U$ exterts a homotopy between $\delta(t)$ and $\eta(t)$ such that $\alpha(t,c_1)=\delta(t)$ and $\alpha(t,d_1)=\eta(t)$.
Similarly, suppose that $\beta(t,h):[a,b]\times[c_2,d_2]\rightarrow U$ exerts a homotopy between $\eta(t)$ and $\sigma(t)$ such that $\beta(t,c_2)=\eta(t)$ and $\beta(t,d_2)=\sigma(t)$. Note that by translation we can pick $c_2$ and $d_2$ for which $\beta(t,h)$ is defined such that $c_1<d_1=c_2<d_2$.
Thus one can define $\gamma(t,h) = \begin{cases} \alpha(t,h) & \text{if $h<d_1$} \\ \beta(t,h) & \text{if $h\geq d_1$} \end{cases}$ with $\gamma:[a,b]\times[c_1,d_2]\rightarrow U$. We see that $\gamma$ is well defined and continuous on the interval and that $\gamma$ has the property that $\gamma(t,c_1)=\delta(t)$ and $\gamma(t,d_2)=\sigma(t)$. Thus $\gamma$ exerets a homotopy between $\delta(t)$ and $\sigma(t)$. Hence, since any path can be parametrised to any any interval it follows that for well defined paths on the same open set $U$ homotopy is a transative property.
If anybody could let me know if this proof is correct, or if it is wrong I would really appreciate it.