After showing that \(\displaystyle B\subseteq A\) you say "In converse let that \(\displaystyle A\subseteq B\). "let" implies that this is part of your hypothesis. It isn't! It is the conclusion that you want to prove. The hypothesis for the converse is that \(\displaystyle B\subseteq A\).
But you may have confused the "converse" with simply proving both parts of "\(\displaystyle A\cap B= B\)". Since "X= Y" implies both "\(\displaystyle X\subseteq Y\)" and "\(\displaystyle Y\subset X\)", you really need to prove four things:
1) if \(\displaystyle A\capeq B= B\) then \(\displaystyle A\subseteq B\).
2) if \(\displaystyle A\capeq B= B\) then \(\displaystyle B\subseteq A\).
3) if \(\displaystyle A= B\) then \(\displaystyle A\cap B\subseteq B\).
4) if \(\displaystyle A= B\) then \(\displaystyle B\subseteq A\cap B\).
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