Proof of monotonocity of preferences Question from Intermediate Microeconomics by Hal Varian:
"We claimed in the text that if preferences were monotonic, then a diagonal line through the origin would intersect each indifference curve exactly once. Can you prove this rigorously?"
What I've done:
Preferences in the indifference curve can be represented using utility function and all the bundles that are indifferent are assigned a single value. And each value of the utility function also represents a fixed point on the diagonal line drawn through the origin. So, if one indifference curve intersects the diagonal line n times (say), then (n-1) points belong to different indifference curves and thus it would mean that the indifference curve in question has intersected another indifference curve, which is fundamentally not possible.
Does it prove the statement satisfactorily?
Let me pick up the discussion in the comments.
Consider any diagonal through the origin. Suppose it intersects some IC more than once. Pick two of these points and call them A and B. Because they are on a straight increasing line, one of the bundles, say B, has more of both good 1 and good 2. By monotonicity, bundle B should be preferred over A, which contradicts the statement that they are on the same IC.

If that is what you mean, your intuition is correct. However, there seems to be a jump in your argument: By saying
"if one indifference curve intersects the diagonal line n times (say), then (n-1) points belong to different indifference curves" you seem to be already assuming that the statement to be shown is correct and the n intersections could not exist in the first place.