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Can anyone explain what is wrong with my reasoning?

Suppose [itex]x = \frac{p}{q}[/itex] and let [itex]x = \sqrt 2 + \sqrt 3 [/itex]. Also, let [itex]a,b,c \in {\Bbb Z}[/itex] and assume [itex]a < xc < b[/itex]. If I show that xc must be an integer, and I know there does not exist c such that [itex]\sqrt 2 c[/itex], or [itex]\sqrt 3 c[/itex] is an integer. Then, [itex]\left( {\sqrt 2 + \sqrt 3 } \right)c[/itex] cannot be an integer, a contradiction.

p and q are integers, where q > 1. I am supposing that [itex]\sqrt 2 + \sqrt 3 = p/q[/itex].

Suppose [itex]x = \frac{p}{q}[/itex] and let [itex]x = \sqrt 2 + \sqrt 3 [/itex]. Also, let [itex]a,b,c \in {\Bbb Z}[/itex] and assume [itex]a < xc < b[/itex]. If I show that xc must be an integer, and I know there does not exist c such that [itex]\sqrt 2 c[/itex], or [itex]\sqrt 3 c[/itex] is an integer. Then, [itex]\left( {\sqrt 2 + \sqrt 3 } \right)c[/itex] cannot be an integer, a contradiction.

p and q are integers, where q > 1. I am supposing that [itex]\sqrt 2 + \sqrt 3 = p/q[/itex].

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