$|a\alpha-b\beta|\leq|a-b|+|\alpha-\beta|$

Question

It may be the case that this is extremely trivial to prove, but I just cannot see it. I want to prove this:

$|a\alpha-b\beta|\leq|a-b|+|\alpha-\beta|$

for $0\leq a,\alpha,b,\beta\leq1$

I think it is also the case that

$|a\alpha-b\beta|\leq|a-b + \alpha-\beta|$

I actually THINK these might hold because I cannot provide any counter example.

Any kind of help would be extremely appreciated.

Answer 1

Apply the $\triangle$ inequality: $|a\alpha - b\beta| = |a\alpha - a\beta + a\beta - b\beta|\le |a||\alpha - \beta|+|\beta||a-b|\le |a-b|+|\alpha-\beta|$