Is this true for real numbers?

Question

$|a_{m+1}b_{m+1}+a_{m+2}b_{m+2}...a_nb_n|<|a_{m+1}(b+1)+a_{m+2}(b+1)+...+a_n(b+1)|$

Is this true if we know that $b_n<b+1$ for any $n$? I am not sure, I would appreciate any help. I think it's true from the triangle inequality but i might be wrong.

Answer 1

No, not at all. For instance, $-1<0$, but $|1\cdot (-1)|<|1\cdot 0|$ is false. Or if you want the $b$'s to be nonnegative, a counterexample is $$|(-1)\cdot 2+1\cdot 1|\not<|(-1)\cdot 3+1\cdot 3|.$$