How to make Roger Penrose's proof of $\sqrt{2}$'s irrationality rigorous? Taken from pg-53 of Roger Penrose's road to reality,
Suppose $\sqrt{2}$ is rational, then:
$$ \sqrt{2} = \frac{p}{q}$$
Where $p$ and $q$ are some integers with $ q \neq 0$:
Squaring and rearrangnig:
$$ 2 q^2 = p^2$$
This means $p$ is even, and hence, $p=2p_1$, meaning that:
$$ q^2 = 2p_1^2$$
This means that $q$ is even, we can put $q=2q_1$, where $q_1$ is some integer, meaning that:
$$ 2q_1^2 = p_1^2$$
See that this looks same as the equation we begun with $2q^2 =p^2$, repeating the procedure we done it again we can further find that $p_1$ and $q_1$ is again even writing $p_1 = 2p_2$ and $q_1=2q_2$, after simplifications we will find again:
$$ 2q_2^2 = p_2^2$$
Repeating the procedure many times, we could repeat this as many times as we want , we get a recurrence for $q_i$s as:
$$q_i = 2q_{i+1}$$
Or, $$\frac{q_i}{2} = q_{i+1}$$
Obeying:
$$q>q_1>q_2...$$
Penrose writes:
"..All of these integers being positive. But any decreasing sequence of positive integers must come to an end, contradicting the fact that this sequence is unending. This provides us with a contradiction to what has been supposed , namely that there is a rational number which squares to $2$"
Firstly where was it assumed that the sequence is 'unending' and what exactly does it mean for a sequence to end? I thought sequence could go on forever..
P.S: I understand the square root of two's irrationality proof already, the point is not to prove the statement in itself but understand this particular proof.

Firstly where was it assumed that the sequence is 'unending'
Here: $$q>q_1>q_2>\ldots$$
He means that it is an infinite strictly decreasing sequence: each later term less than the previous one.  This infinite sequence is constructed inductively.  He uses the ellipsis notation to suggest this.
what exactly does it mean for a sequence to end? I thought sequence could go on forever..
Yes, a sequence can go on forever. But an infinite sequence of positive integers cannot strictly decrease forever. That is what the well-ordering principle says.
Compare this to the case of rational numbers, where you can have $1/2 > 1/4 > 1/8 > \ldots > 1/2^n > \ldots $ forever.
