Is a certain restriction of an open map open? Let $p:X\rightarrow Y$ be an open map and let $A$ be a subspace of $X$. Then, is it true that $p|_A:A\rightarrow p(A)$ is open?

I think so, but am struggling to show it. 

My thoughts: Let $O$ be an open set in $A$. Then, since $A$ has the subspace topology, there exists a $U$ open in $X$ such that $U\cap A=O$. But then, $p(O)=p(U\cap A)$. Yet, I only know that $p(O)=p(U\cap A)\subseteq p(U)\cap p(A)$, which doesn't really help.

If I assume $p$ is also a quotient map, does this help? Or can I do it without that assumption?
The result is not true. See here.

I am not flagging this as duplicate because of the second part of your question. 

Also not true for quotient map. Take $p: \mathbb{R} \rightarrow S^1$ (usual topology on the reals, quotient by integers). Then let A be [0.1,0.2]. Then $p(A) $ is not open in $S^1$.
Let X be the real line, Y its quotient by the integers and p be the quotient map. Let A=[0, $\infty)$ and consider the open set [0, 1) in A. Its image is not open in p(A) (which is the whole of Y).