First, I'm glad to see that your gender hasn't changed.
Second, BMI is a pretty worthless method of tracking progress for people who lift weights.
Third, what method is your trainer using to measure your bodyfat? Caliper pinching, handheld electronic device, etc.?
Lastly, in six months you've gained about 10 pounds of total bodyweight? That's slow progress. How have you been eating? What, exactly, did you eat yesterday?
I'm not trying to blatantly disagree with your trainer, but a dude as underweight as you are absolutely does not need one entire training day, a third of your training week, just for arms.
At the rate you've been progressing, 10 weeks is a drop in the bucket and I wouldn't expect any significant size changes at all. Again, consider what it took you the last six months to accomplish. And now we're trying to plan for the next two and a half? It's too short, man.
You need to realize that at your height, most guys won't look "built" until they're well over 200 pounds. Everything until then is just bringing you up to a certain base point. Of course everyone needs to start somewhere, I'm not trying to be a buzzkill, but there's no need to take things slower than necessary. I just have a feeling either you or your trainer are playing things a little too conservative.
Try setting a longer-term goal, like where you'd like to be a year or two from now both size and strength-wise. And then track back, breaking it into shorter goals to reach that point.
Also, since you did go through the expense of getting an actual trainer, have you asked them these same questions (RE: expectations and goals)? What was their answer?