TITLE: Induction Proof:
QUESTION [0 upvotes]: I am stuck on this problem. I understand the fundamentals of induction proofs, but I am unfamiliar with induction on two variables. Here's the prompt:
Prove that for every positive integer $k$, the following is true: For every real number $r > 0$, there are finitely many solutions to $\frac{1}{n_1}+\frac{1}{n_2}+...+ \frac{1}{n_k} = r$. In other words, there exists some number $m$ (that depends on $k$ and $r$ such that there are at most $m$ ways of choosing a positive integer $n_1$, and a (possibly different) positive integer $n_2$, etc., that satisfy the equation.
I have no clue where to start, especially how to reference $m$ in terms of $k$ and $r$. All I know is we cannot induct on $r$ because it is not a natural number.

REPLY [0 votes]: Just because multiple variables are involved does not mean that you have to do induction on all those variables:  often you just induce over one variable, while you take the other variables as arbitrary (i.e you basically do a universal proof over all the other variables).  
Of course, sometimes you don't need to use induction at all, and can do a universal proof for all variables ... or maybe you do induction over 2 out of the 5 variables or ...
In general: for each variable you have a choice of induction or universal proof, though you are right that induction only works for a recursively defined set of objects like the natural numbers.
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