Kagome is lost to him, and this makes him careless, makes him...well, kind of stupid. Sango and Miroku have both tried to reason with him, Kaede has tried to calm him down, even Shippo has stepped forth to try and get through to him.
Even Koga has tried. Come on, you stupid mutt, do you think this is what she'd want? Moron! You're gonna get yourself killed! And what happens if she does manage to get back, huh? Find out you're dead, and break her heart all over again? Idiot half-breed!
But he doesn't listen, and instead does foolish, reckless things, such as attack larger, more powerful demons head-on, occasionally without even attempting to unsheathe Tetsusaiga, alienating his friends, and seeking out Sesshomaru to pick a fight.
Sesshomaru recognizes his brother's self-destructive impulses for what they are, and although there was a time, not so long ago, when he would have taken the opportunity to slay him and claim Tetsuseiga, he now realizes it would prove nothing, as he can't even wield the sword.
It reaches the point where Sesshomaru will simply turn his back on his brother and walk away, refusing to battle him.
What's wrong, Sesshomaru? Afraid? Or am I makin' it too easy for you?
His brother turns back to face him, looks at him for a long moment. You are pathetic. How such a weak half-breed shares my blood is a mystery to me. How you have even survived is beyond my comprehension.
Inuyasha sneers. Don't sound too hard to me.
You possess Tetsuseiga, yet you do not welcome it. You had possessed one who was so close to you, yet you pushed her away. And now you are feeling sorry for yourself. Thrusting yourself into battle when you do not even know that you are in too deep. I will not waste my time on such a pathetic excuse for a demon.
Inuyasha is furious. What the hell do you know about it, anyway? You wouldn't know...know love if it walked up and bit you on the ass, so don't try to tell me what I feel! You're a gods-damned hypocrite, that's what you are!
That may be so, but I am not the one who is giving up, when all you have to do is wait. With that, Sesshomaru turns and starts walking again.
Inuyasha stares after him, completely stunned. What the hell does that mean?
Sesshomaru pauses, but does not turn back. You are the son of the Inu no Taisho, a half-demon. What makes you think that your life is as short as that of these humans?
I'm not immortal!
No. But you are not going to die in the near future. You have more time than you are willing to give yourself.
And then he walks away, and even though Inuyasha calls his name, he does not turn back.
End