Ytukyg

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 $$lim_{ntoinfty}{a_n} in [0,1).$$ 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.

Your condition is sufficient, but not necessary. Take, for instance$$a_n=begin{cases}frac12&text{ if $n$ is odd}\0&text{ otherwise.}end{cases}$$Then $lim_{ntoinfty}a_n$ doesn't exist, but $lim_{ntoinfty}lfloor anrfloor=0$.

A condition which is both necessary and sufficient is that $ngg1implies a_nin[0,1)$. Can you prove it?

$(lfloor a_n rfloor)$ is a stationnary sequence.

If $lfloor a_nrfloor$ goes to zero, then for large enough $n$,

$$-frac 12<lfloor a_nrfloor <frac 12$$

thus
$$lfloor a_nrfloor=0$$

and

$$0le a_n<1$$