If I start a proof by saying:
Assume $a^3=b$
and end up proving something with it, will the proof hold for when $a^3\neq b$? Would that be a valid proof for all $a$ and $b$ baring any other domain issues?
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"Assuming $a^3=b$. Then clearly $a^3+1=b^3+1$." Do you we should be able to conclude "$a^3+1=b^3+1$" for all $a, b$? – Noah Schweber Sep 19 '17 at 02:39
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In general, no. For instance, if one assumes $x=2y$, one can prove $x^2=4y^2$, and the result is true in other cases (like if $x=-2y$), but it is not true in general (take $x=y=1$, for example).
Carl Schildkraut
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