What you should observe here is that . That's what shows that a ~ a.

That actually doesn't follow.

What you need to do is show that starting with ,

you can show that there's some j such that . What could j be?

Working backwards, if both those hold, then: , and ,

so , so , so

So find n such that and then will imply .

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For the transitive: Those aren't "the same i".

From a~b and b~c you should conclude that:

After that consider .