Is there a mistake in this proof? 
Proposition 1: The number of occurrences of an event within a unit of
  time has a Poisson distribution with parameter  $\lambda $ if and only
  if the time elapsed between two successive occurrences of the event
  has an exponential distribution with parameter  $\lambda $ and it is
  independent of previous occurrences.

Source
Now, if we look in the proof, it starts with:

Since  $X \ge x$ if and only if $\tau_1 + \tau_2 + \cdots + \tau_x \le
 1$ (convince yourself of this fact), the proposition is true if and
  only if:
$P(X\ge x)=P(\tau_1 + \tau_2 + \cdots + \tau_x \le 1)$ for any $x \in
 R_X$.

'The proposition is true if and only if' means that implication works in both ways. So we have two implications:
1) Proposition is true $\implies$ $P(X\ge x)=P(\tau_1 + \tau_2 + \cdots + \tau_x \le 1)$
2) $P(X\ge x)=P(\tau_1 + \tau_2 + \cdots + \tau_x \le 1)$ $\implies$ Proposition is true
The author doesn't say why two implications above are true. Could anyone explain that? It looks like the author proves the equality $P(X\ge x)=P(\tau_1 + \tau_2 + \cdots + \tau_x \le 1)$, but doesn't show why it implies Proposition 1. Therefore, the proof doesn't prove what it's claims to prove.
Note: the source has been updated and now takes into account the comments provided by the users on this page.

Assumptions: $X$ is the number of arrivals in the time interval $(0,1]$
(note the endpoints), and the time of
the $n$-th arrival is $\tau_1+\tau_2+\cdots+\tau_n$. Note that $\tau_k$ is
the $k$-th interarrival time.
Then, if the event $\{X \geq n\}$ has occurred, then there are at least $n$ arrivals in $(0,1]$, and so we are assured that the 
$n$-th arrival
must have occurred no later than at time $t = 1$. Thus, we
we can be sure that $\tau_1+\tau_2+\cdots+\tau_n \leq 1$.
On the other hand, if $\tau_1+\tau_2+\cdots+\tau_n \leq 1$, then
we know that the $n$-th arrival occurred at time $t \leq 1$, and
so we are guaranteed that at least $n$ arrivals have occurred
in the time interval $(0,1]$, that is, the event $\{X \geq n\}$
has occurred.
Of course, none of this addresses the issue of the independence
of the $\tau_k$, their exponential distribution, or the
Poissonity of $X$.
