No jumping here, and you're right to ask that question. But it's probably true. If not, he'd have to forge it, and what are the chances he could do so unobserved? Plus, this is something I wouldn't put past members of the Left, until recently (before it was taken down with extreme prejudice by hackers) there was a website called "oppose the troops" that made fun of wounded and dead soldiers. Heck, for all we know the Code Pink jerks who protest out in front of Walter Reed might have sent it.
But here's another angle: Even if we were being conned, the net result of the con is that a response of thousands of letters would be a boost to troop morale.
I'm sure it's true, and if I'm wrong, there's little or no downside.