The phrase "continue this process" in proof statements

It's a (justified but) lazy way of concealing a proof by induction.

Generally, proofs where you 'continue the process' typically involve some variable $n$ that decreases each time you do something; when $n$ eventually reaches $0$, the result is true for some obvious reason.

For instance, in your example, the result being proved is: for each $n \ge 0$, if there is a path from $u$ to $v$ that contains exactly $n$ cycles, then there is a simple path from $a$ to $b$.

• The base case says "if there is a path from $a$ to $b$ that contains $0$ cycles, then there is a simple path from $a$ to $b$". This is trivial, since a simple path is a path with no cycles.

• In the induction step, you're assuming for some $n \ge 0$ that if a path from $a$ to $b$ contains $n$ cycles, then there is a simple path from $a$ to $b$. Then if a path has $n+1$ cycles, remove one of them; et voilà, you have a path with $n$ cycles, so there exists a simple path by the induction hypothesis.

This then proves the result you need, since any path can only have a finite number of cycles in it.

If you want to be formal: Yes, there’s always an implicit induction behind statements like this. However, mental reasoning is not reflected by formal mathematics and such statements are meant to aid your mental reasoning and highly increase the readability of proofs over formal ones.

If you start mathematics, it’s a good exercise to formalize these hidden inductions every now and then, especially if you feel not fully convinced by the mental images conjured by the informal arguments. Not every plausible idea holds true in real life, so you need to develop a good intuition about these things to minimize the risk of falling into a trap. You do this by exercising your formal skills.