Upper bound of function explanation .

Question

In a paper about numerical SDE's the author writes that for a certain function $b(t,x)$ , we define $$ b_n(t,x) = \frac{1}{1 + n^{-a} \left | b(t,x) \right | }b(t,x) $$

for $ x\in \mathbb{R}^d, t \in[0,T]$.

Then he writes that $$ \left | b_n(t,x) \right | \leq min(n^a, \left | b(t,x) \right |). $$ I know that this holds, but I can't see the intuition behind it. Every answer is welcome.Thanks.

Answer 1

When you set $A=n^a>0$ and $B=|b(t,x)|\ge 0$, and additionally apply $$ AB = \min(A,B)·\max(A,B)\\ \max(A,B)\le A+B $$ then the quantity in question can be written as and bounded as $$ |b_n(t,x)|=\frac{AB}{A+B}\le\frac{\min(A,B)·(A+B)}{A+B}=\min(A,B), $$ as was claimed.