Question

There are infinitely many stations on a train route. Sup-

pose that the train stops at the first station and suppose that if the train stops at a station, then it stops at the next station. Show that the train stops at all stations.

Short answer

Examples of Hypothesis In the conditional, "If all four sides of a quadrilateral measure the same, then the quadrilateral is a square" the hypothesis is "all four sides of a quadrilateral measure the same".

Step-by-step solution

Step 2

Let p(k) be the hypothesis that the train stops at station k.

Base case: P(1) is true because we are told that the train stops at the first station.

Induction step: Assume that the train stops at station k - 1 that is P ( k - 1 ) , is true.

Now, we are told to assume that, if the train stops at a station, then it stops at the next station. Since we assumed that it stopped at station k - 1 (by induction hypothesis), it means that it will stop at station k. Thus P (K) , is true.

Then, by induction, P(k) is true for all k and, therefore, the train stops at all stations.