Weak Induction Proof - Variables

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Can someone help me understand this problem? Why is it necessary in step 2 to set n=m? https://i.stack.imgur.com/xqWWb.jpg

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They want to show that if the statement is true for $n=m$ then it is also true for $n=m+1$

Some authors use if true for $k$ then it is true for $k+1$

Some authors use if it is true for $n$ then it is true for $n+1$

It is a matter of taste and they are all equivalent.