Is the Proof in Dummit and Foote's Chinese Remainder Theorem Inductive?

[icantadd]

In Dummit and Foote on pages 265-266, a proof is given of the Chinese Remainder Theorem. They claim to proceed by induction, but I cannot see where the induction hypothesis is used.

It seems that they could proved the statement for k=2, and then reduced the statement for k>2 to k=2. Is this induction?

Thank you for your help.

 

[ramsey2879]

In Dummit and Foote on pages 265-266, a proof is given of the Chinese Remainder Theorem. They claim to proceed by induction, but I cannot see where the induction hypothesis is used.

It seems that they could proved the statement for k=2, and then reduced the statement for k>2 to k=2. Is this induction?

Thank you for your help.

I don't have a definite answer, but my feeling is that if your proof for values of n greater than 2 is base upon a showing that it is dependent upon the validity of the case for n = 2 then it is a form of induction.