Nope, I read your first sentence and stopped. What you said is not true! Dividing by t will not necessarily make the fraction smaller.
If k>0, consider k/1=k, k/(1/2)=2k>k, k/(1/3)=3k>k, k/(1/4)=4k>k...
I am dividing k by various numbers and the fraction are NOT smaller than k. Sorry. Try again