For example, in proving the following lemma
If $n\in\mathbb{N}$ then $n+1=1+n.$
Little gave in his book entitled THE NUMBER SYSTEMS OF ANALYSIS the following proof: Clearly the lemma holds if $n=1.$ Assume as an inductive hypothesis that $n+1=1+n$ for some $n\in\mathbb{N}.$ Then \begin{align*} (n+1)+1&=(1+n)+1\\ &=1+(n+1) \end{align*} by associativity. Thus the lemma follows by induction. $\Box$
My question is: Is it true that the italicized "some" equals "any"? That is, the sentence
\begin{gather*}
\text{Assume as an inductive hypothesis that $n+1=1+n$ for some $n\in\mathbb{N}$}
\end{gather*}
says the same thing as the sentence
\begin{gather*}
\text{Assume as an inductive hypothesis that $n+1=1+n$ for any $n\in\mathbb{N}$?}
\end{gather*}
Many thanks.
No. If you use the second statement, then you implied that the proposition is right, which is the thing that you are proving. This is cycle reasoning.
What i am saying is, the 2nd statement is just the proposition above. You cannot assume that.