I'm new in that stuff in proving things. I'm always confused about when is a really valid reasoning. Then is this property, i appreciate any help.
Prove, for any real x:
x - 1 < $\lfloor$x$\rfloor$ $\leq$ x $\leq$ $\lceil$x$\rceil$ < x + 1
by definition:
$\lfloor$x$\rfloor$ = n $\iff$ n $\leq$ x < n + 1
$\lceil$x$\rceil$ = n' $\iff$ n' - 1 < x $\leq$ n'
so, $\lfloor$x$\rfloor$ $\leq$ x $\leq$ $\lceil$x$\rceil$ , because x $\leq$ n $\land$ x $\leq$ n'
( $\lfloor$x$\rfloor$ + 1 + (-1) > x + (-1) ) $\land$ ( $\lceil$x$\rceil$ - 1 + (1) < x + (1) )
($\lfloor$x$\rfloor$ > x - 1) $\land$ ($\lceil$x$\rceil$ < x + 1)
Thereof, x - 1 < $\lfloor$x$\rfloor$ $\leq$ x $\leq$ $\lceil$x$\rceil$ < x + 1