I am having trouble with the following question:
Let $M=\{y\in\mathbb{R}\colon y=-4x+10 \text{ for some } x\in(6,11)\}.$ Show that the least upper bound and greatest lower bound of $M$ are $-14$ and $-34$, respectively.
I know that $M = \{-26, -22, -18\}$, but I don't really know which set $M$ is a subset of. Maybe that's my problem.
As mentioned by Henry, $x$ is any real number between $6$ and $11$, not just one of the integers $\{7,8,9,10\}$. For example, $x$ can be $7.5$, or even $7+\pi/2$.
Hint: show that finding the least upper bound is equivalent to finding $\max_{x\in[6,11]} \{-4x+10\}$. Perhaps the continuity of $x\mapsto -4x+10$ is useful here.
You can use the same trick for the greatest lower bound.