This was an exercise in my lecture notes for which no answer was provided, so I seek verification on whether my proof is correct.
Prove that if
1. $P(n_0)$ is true for some $n_0 \in \mathbb N$, and if
2. $\forall k \in \mathbb N : k \ge n_0 : P(n_0) \land P(n_0 + 1) \land \ldots P(k - 1) \land P(k)$ true $\Rightarrow P(k + 1)$ is true,
then $P(n)$ is true $\forall n \in \mathbb N, n \ge n_0$.
Proof
Let $S = \{x \in \mathbb N: x \ge n_0: P(x)$ is false$\}$.
If $S$ is non-empty, then S contains a minimum element by the well-ordering principle of $\mathbb N$, call it $m$, where clearly $m \ne n_0$.
Then $P(n_0), P(n_0 + 1), \ldots, P(m - 2), P(m-1)$ cannot be false, which implies $P(m)$ is true; a contradiction.
Therefore $S$ = $\emptyset$ and $P(n)$ is true $\forall n \ge n_0$.
Yes, this proof is correct. Nice work!