bounding a certain quantity

Question

Is there a way to bound the following quantity $$ |\sqrt{a_n - b_n} - \sqrt{a-b}| $$ with an expression composed of the expressions $a_n-a$ and $b_n -b$?

Answer 1

It is bounded by $\frac {|a_n-a|+|b_b-a|} {\sqrt {|a-b|}}$. You get this multiplying numerator and denominator by $\sqrt{a_n-b_n}+\sqrt {a-b}$.