Let $(a_n)_{n=1}^\infty$ be a real sequence. Find a necessary and sufficient condition for $(a_n)$ so $(\lfloor a_n \rfloor)_{n=1}^\infty$ converges to $0$.
Hi everyone. I am trying to brush up on calculus. I believe the necessary and sufficient condition would be $$\lim_{n\to\infty}{a_n} \in [0,1).$$ It's clear for me why this is true (or could be) but I need to prove this using the definition of a limit. I've tried some tricks with the triangle inequality and the properties of the floor function, but for some reason I can't prove this properly.
I would love to hear your thoughts.
Best Answer
$(\lfloor a_n \rfloor)$ is a stationnary sequence.
If $\lfloor a_n\rfloor$ goes to zero, then for large enough $n$,
$$-\frac 12<\lfloor a_n\rfloor <\frac 12$$
thus $$\lfloor a_n\rfloor=0$$
and
$$0\le a_n<1$$