Flawed proof that all positive integers are equal

Are you familiar with the proof that all horses are the same color? Try googling that phrase, it fails in a similar (but not identical) way. Try working through the case $k=1$ of your problem, and think about what might go wrong with $x$ and $y$ when you try to apply your base case.

Try $x=1$, $y=2$. Note that $x-1$ is not a positive integer.

How to find $ \lim_{(x,y)\to(0,0)} \frac{\sin^2 (x-y)}{|x|+|y|} $?

How to calculate no. of binary strings containg substring “00”? [duplicate]

Example to the statement that $a_{n+1} - a_n \rightarrow 0$ as $n \rightarrow \infty$ does not imply that sequence $a_n$ converges. [duplicate]

Is $\bar X$ a minimum variance unbiased estimator of $\theta$ in an exponential distribution?

Find the eigenvalues and their multiplicities of a special matrix

Find the tangent lines to the graph of $x^2+4y^2 = 36$ that go through the point $P=(12,3)$

A finite $p$-group cannot be simple unless it has order $p$

Simple binomial theorem proof: $\sum_{j=0}^{k} \binom{a+j}j = \binom{a+k+1}k$

Prove that $(x+y+z)^2(yz+zx+xy)^2 \leq 3(y^2+yz+z^2)(z^2+zx+x^2)(x^2+xy+y^2)$

Why I can't login to my Mac without typing in my password sometimes?

False insufficient-disk-space alert