St. Augustine writes (De Civ. Dei i, 20): “Hence it follows that the words ‘Thou shalt not kill’ refer to the killing of a man—not another man; therefore, not even thyself. For he who kills himself, kills nothing else than a man.”
To kill oneself is to kill a man; to kill a man is wrong; so, to kill oneself is wrong. Suicide is homicide; homicide is wrong; ergo, etc. Tightening up the argument:
1) Every intentional killing of a human being is morally wrong.
2) Every act of suicide is the intentional killing of a human being.
Therefore
3) Every act of suicide is morally wrong.
The syllogism is valid, but the major is not credible. Counterexamples in decreasing order of plausibility: just war, capital punishment, self-defense, abortion in some cases, and, of course, suicide!
Note that (1) cannot be supported from the "Thou shalt not kill" of the Decalogue. As Paul Ludwig Landsberg correctly comments, "The Christian tradition, apart from a few sects, has always allowed two important exceptions: [just] war and capital punishment." (The Experience of Death, tr. Rowland, New York: Arno Press, 1977, p. 78) I would add that the allowance is eminently reasonable.
How could suicide count as a counterexample to (1)? Well, as Landsberg points out, killing oneself and killing another are very different. (79) As I would put it, in a case of rational suicide such as the case my reader proposes, one kills oneself out of loving concern for oneself whereas the killing of another is typically, though not always, a hostile and hateful act.
Although Augustine's argument cannot be dismissed out of hand, it is not rationally compelling.