And yeah non-uniform convergence works. I would've just used this guy:

\[ f(x,y) = \frac{x^2}{x^2+y^2}\text{ with limits as }x\to 0, \, y\to 0 \]

Alternatively just use the example in the previous question:

\[ f(x,y) = 2\arctan \left(e^x \tan \frac{y}{2} \right)\text{ with limits as }y\to 0, x\to \infty\]

High school questions: Well firstly there's still this one which I will leave alone.

1. Prove the cosine rule in trigonometry using properties of the vector dot product

Otherwise this is a relatively straightforward matrix question provided you know what induction is.

\[ \text{Let }D\text{ be a diagonal matrix. Use mathematical induction to prove that for }n\in \mathbb{Z}_+\\ \text{the terms of }D^n\text{ are just the individual components raised to the }n\text{-th power.} \]

First year university: Another straightforward one.

\[ \text{Let }A\text{ be a square matrix with eigenvalue }\lambda \text{ and eigenvector }\mathbf{v}.\\ \text{Prove by induction that }\mathbf{v}\text{ is an eigenvalue of }A^n\text{ where }n \in \mathbb{Z}_+\\ \text{and find the corresponding eigenvalue.}\]

First year university:

\[ \text{Show that if }\lambda\text{ is an eigenvalue of }A\in \mathbb{R}^{n\times n}\\ \text{then it is also an eigenvalue of }A^T\]