Say we want to prove $\lfloor nx\rfloor\leq n+n\lfloor x\rfloor$, but instead we prove, $\lfloor nx\rfloor<n+n\lfloor x\rfloor$, are we done?

Question

Just a bit of a silly question:

Say we want to prove:

$\lfloor nx \rfloor \leq n + n \lfloor x \rfloor$, but instead we prove,

$\lfloor nx \rfloor < n + n\lfloor x \rfloor$, are we done?

Answer 1

When we write $x\le y$ for real numbers $x$ and $y$, we mean "either $x<y$ or $x=y$". Thus, if $x<y$, then certainly "either $x<y$ or $x=y$" is true, whence $x\le y$.