Equivalent statements of Orthogonal Projections

I am trying to prove the following equivalent statements.

Let $P$ and $Q$ be two orthogonal Projections defined on space $H$. Then the following are equivalent:

1. $PQ=QP=P$;
2. $Im(P) \subset Im(Q)$;
3. $ker(Q) \subset ker(P)$;
4. $(Px,x) \leq (Qx,x), \forall x \in H$;
5. $|| Px|| \leq ||Qx||, \forall x \in H$.

I have managed to prove $(1) \implies (2)$ , $(2)\implies (3)$ and $(4) \implies (5)$. I am stuck in the remaining implications, i.e $(3) \implies (4)$ and $(5) \implies (1)$. Please if anyone can help?

Many Thanks

For $(3) \implies (4)$, I suggest you first prove $(3)\implies (1)$. Note then that $Q-P$ is a projection since $(1) $ obviously implies $$(Q-P)^2 = Q^2 - QP -PQ +P^2 = Q- P -P + P = Q-P.$$ Since $Q-P$ is a projection, it is positive and thus $Q\ge P$ which is equivalent to $(4)$.

For $(5)\implies (1)$, note that $$\|(P-PQ)x\| = \|P(1-Q)x\| \le \|Q(1-Q)x\| = 0$$ so $P = PQ$. Taking adjoints, $P = QP$. Thus $(1)$ holds.