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anggar20
anggar20
11.02.2020 • 
Mathematics

Identify all mistakes made in the following proof that for each integer n, there is an integer k such that n < k < n+2.

Suppose n is an arbitrary integer. Therefore, k = n + 1 for all integers n. This means that n < n + 1 < n + 2. This proves that an integer k exists.

ASuppose n is an arbitrary integer. Therefore, k = n + 1

The choice of k = n + 1 does not follow from the fact that n is an integer. The inappropriate word "Therefore" should be replaced by "Pick", "Select", "Choose" or other words to that effect.

BTherefore, k = n + 1 for all integers n.

After we decided that n is some integer, n is just that - an integer. It's no longer a free variable. It makes no sense to apply universal quantification to it, again just like it makes no sense to say " 2 + 3 = 5 for all integers 5".

CTherefore, k = n + 1 for all integers n. This means that n < n + 1 < n + 2.

It makes no sense to draw a conclusion from a variable definition that does not reference the variable. n < n + 1 < n + 2 is true for all n regardless of what k is. The proper conclusion to be drawn from the definition of k is that n < k < n+2.

D This proves that an integer k exists.

We know that integers exist. If we're going to make a concluding statement that summarizes the theorem we have proved, we must quote the theorem correctly, not a caricature of our theorem. A correct concluding statement would be: this proves that for every integer n, an integer k exists such that n < k < n+2.

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