I read this sentence about the sum of squared error SSE: "The square ensures the error is always positive and larger errors are penalized more than smaller errors". I want to understand what is meant by "larger errors are penalized more than smaller errors", and how to prove that?
Plot $x$ against $x^2$. Smaller values squared change less, then larger values squared. The steepness of the slope of $x^2$ increases as $|x|$ grows. So if you square small errors, the penalty is smaller, then if you squared large errors. There is nothing to prove in here, it is just how the square function works.