Just share a minor point in the presentation of M.I.
To prove that a proposition P(n) is true for all positive integers n by using M.I.
We need ‘4’ steps, namely
Just want to say something about [Assuming].
The textbook gives a good presentation like
“For any positive integer k, assume that P(k) is true,…"
“For any positive integer k" is NOT describing “assume that P(k) is true".
“For any positive integer k" is describing the fact that
“assume that P(k) is true, then P(k + 1) is also true".
or I prefer writing
“if P(k) is true, then P(k + 1) is true".
Urm, this is the principle of M.I..
Students always puzzled that why we CAN assume P(k) is true?
But actually, we are NOT merely “assume P(k) is true", but we are checking whether “" is true or not.
(Note that I used instead of here.)
It is good to mention “For any positive integer k", because we actually need to describe clearly that what k is, as well as, we need to ensure, “For any positive integer k, “. Hence, it is actually not very good in writing something like
“Assume P(k) is true for some positive integer k, …"
P(k) is true for some k? The principle of M.I. may not be applicable.
Of course, it is WRONG to write
“Assume P(k) is true for any positive integer k."
Because, in this case, “for any positive integer k" is really describing the fact that “P(k) is true", and nothing we need to prove at all.
But, to avoid making confusion, I require student to write
“Assume P(k) is true."
Urm, this is NOT a clear expression. k is not defined clearly. But, at least, it is a ‘safe’ way of presentation in the public examination.
1. 「數學歸納法」還是「數學演繹法」？ (written by Mr. Leung)