Let $c:[a,b]\rar M$ be a smooth curve in the Minkowski space $M$ such that for all $t\in[a,b]$: $c^\prime(t)$ is time-like and future pointing. Then
$$
T(c)\colon=\int_a^b\Vert c^\prime(t)\Vert\,dt
$$
is the proper time of the curve. Show that for any smooth curve $c:[0,1]\rar M$ such that $c^\prime(t)$ is time-like and future pointing and $c(0)=0$ and $c(1)=(1,0,\ldots,0)$: $T(c)\leq1$.
Since $c$ is time-like and future pointing, we have for all $t$: $\la c^\prime(t),E^0\ra < 0$ and
$$
c^\prime(t)=-\la c^\prime(t),E^0\ra E^0+k(t)
$$
where $k(t)\perp E^0$, i.e. $k(t)$ is a space-like vector and thus
$$
\tnorm{c^\prime}
=\sqrt{-\la c^\prime,c^\prime\ra}
=\sqrt{\la c^\prime,E^0\ra^2-\la k,k\ra}~.
$$
Since $\la c^\prime,E^0\ra\leq0$ and $\la k,k\ra\geq0$, we obtain:
\begin{eqnarray*}
T(c)
&=&\int_0^1\tnorm{c^\prime(t)}\,dt
=\int_0^1\sqrt{\la c^\prime(t),E^0\ra^2-\la k(t),k(t)\ra}\,dt\\
&\leq&\int_0^1\sqrt{\la c^\prime(t),E^0\ra^2}\,dt
=\int_0^1-\la c^\prime(t),E^0\ra\,dt\\
&=&\int_0^1 c_0^\prime(t)\,dt
=c_0(1)-c_0(0)
=1~.
\end{eqnarray*}