If the product $x_1x_2 ··· x_n$ is zero, then $x_1$ and $x_n$ are zero, is this statement true

Asked Jan 06 '21 at 17:41

Active Jan 06 '21 at 17:42

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Let $n$ be a natural number greater 1 and let $x_1, x_2,..., x_n$ be real numbers. Consider the following statement.

If the product $x_1x_2 ··· x_n$ is zero, then $x_1$ and $x_n$ are zero

I'm pretty sure that this statement is not true, since as long as one of these $x_i$ for which $1<i<n$ is zero then this would be some argument to suggest that the statement is false, am I correct in stating this?

edited Jan 06 '21 at 17:42

Semiclassical

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asked Jan 06 '21 at 17:41

mathyman

1

A single counterexample, like $1\times 0\times 1$ is all you need to disprove the claim. – lulu Jan 06 '21 at 17:42
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Correct: A counterexample, as you indicate, is any sequence such that at $x_1,x_n$ are nonzero but one other $x_i$ is zero. – Semiclassical Jan 06 '21 at 17:43
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In fact $1\le i\le n$ is as good. – Jan 06 '21 at 17:46

If the product $x_1x_2 ··· x_n$ is zero, then $x_1$ and $x_n$ are zero, is this statement true

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