Let $p = p(n,m)$ be a property that depends on $n,m \in \mathbb{N}$. If we want to show that it is true using ordinary induction, we see that $p$ holds for all $n,m \geq 1$ iff the property $$Q(n) := \left\{p(n,m) : m \geq 1\right\}$$ holds for all $n \geq 1$, and therefore we can apply ordinary induction on $n$ to show that $p$ is true. My questions are the following ones:
1) Is this approach above as simple as it looks like or am I skipping some logical issues?
2) Consider the case when $p$ depends on an uncountable number of parameters, each parameter being a a natural number. For example, if $p = p(\mathbf{x})$, where $\mathbf{x} \in \mathbb{N}^{\mathbb{R}}$. What can we say in this case?