If $P$ is a poset such that $\forall Q \subset P$ has infimum, then $P$ is a complete lattice

Question

If $P$ is a poset such that $\forall Q \subset P$ has infimum, then $P$ is a complete lattice

My try

Let $Q \subset P$ and let be $Q' \subset P$ the set of all uppers bounds of $Q$

then $Q'$ has infimum $\wedge Q'$ then $\forall q \in Q$ $q\leq \wedge Q'$

and for all $q' \in Q'$ $q\leq q'$ and $\wedge Q' \leq q'$

then $\wedge Q' $ is the supremum of $Q$ and then $P$ is a complete lattice

Answer 1

The statement $$ \forall q \in Q, q \leq \bigwedge Q’ \tag1 $$ in your second line is true, but you need to explain a little bit more.

Can you see why $$ \forall q \in Q \, \forall q’ \in Q’, q \leq q’ $$ is true? Can you see why this implies $(1)$?

Also, note that $(1)$ means $\bigwedge Q’$ is an element of $Q’$, so, $\bigwedge Q’$ is the least element of $Q’$. In other words, $\bigwedge Q’$ is the least upper bound (a.k.a. supremum) of $Q$.