Is $47$ the largest whole number?

Question

I was shown the following proof by contradiction, which somehow shows that $47$ is the largest whole number. Obviously this is not true, but I am not exactly sure where the proof goes wrong. Here it is:

Assume $47$ is not the largest whole number. Let $n$ be the largest whole number. $$n > 47 \implies n-47>0 \implies (n-47)^2 > 0 \implies n^2 - 94n + 2209 > 0 \\ \implies n^2-93n+2209>n$$

Since we defined $n$ to be the largest whole number, the inequality $n^2-93n+2209>n$ is false. Therefore we have reached a contradiction thus proving $47$ to be the largest number.

My thoughts: It seems like this "proof" can be done with any number (ie $47$ does not serve a special purpose in the proof). This leads me to believe that defining $n$ be the largest number is the mistake. Is my evaluation correct?

Answer 1

"Let $n$ be the largest whole number ..."

"$47$ is not the largest number" isn't the only assumption you're making in the proof. When you bring $n$ into the picture, you're implicitly assuming that such an $n$ exists in the first place - that is, that there is a largest whole number.

Answer 2

The assumption " Let n be the largest whole number" implies the existence of a largest whole number which is proved to be false. From a false predicate you can come to any conclusion whatsoever.

All you are proving is that " If there exists a largest whole number then that largest whole number is 47 or any other number". Therefore you implicitly proved that there is no largest whole number.