Weierstrass... thing There is in my maths text-book this property/theorem given under the name of Weierstrass property/theorem: Let $ (a_n) $ be a sequence of real numbers. a)If $ (a_n) $ is monotonically increasing and has an upper bound, then $ (a_n) $ is convergent. b)If $ (a_n) $ is monotonically decreasing and has a lower bound, then $ (a_n) $ is convergent. When searching on Google, I can't find anything related to this, so my first question is: What is the name of this thing? What I want to do is to attempt to prove this property/theorem, I would like some hints on what to start with and what should I end with. Thanks!

HINT: Use the fact that $\Bbb R$ has the least upper bound property: if $\varnothing\
e A\subseteq\Bbb R$, and $A$ is bounded above, then $A$ has a least upper bound. If your increasing sequence is $\langle x_n:n\in\Bbb N\rangle$, let $x$ be the least upper bound of $\\{x_n:n\in\Bbb N\\}$, and show using the definitions of _least upper bound_ and _limit of a sequence_ that $x=\lim\limits_{n\to\infty}x_n$.

The proof for the other result is similar. Alternatively, if $\langle x_n:n\in\Bbb N\rangle$ is decreasing and bounded below, then $\langle-x_n:n\in\Bbb N\rangle$ is increasing and bounded above, so by the first result it converges to some $x$. Now show that $\langle x_n:n\in\Bbb N\rangle$ converges to $-x$.

Forgot to say: in my experience it’s usually called the _monotone convergence theorem_.