Proof that &#x230A; &#x2212;<!-- − --> x &#x230B; </mrow> = &#x221

Simone Werner

Simone Werner

Answered question

2022-05-21

Proof that x = x
I wanted to ask if this kind of reasoning for proving the result in the title could be considered correct:
We know that: x = n if and only if n 1 < x n
Then x = n if and only if n 1 < x n
Then multiplying by -1 the formula n 1 < x n we get n + 1 > x n, inverting the sign.
But n + 1 > x n is equivalent to n x < n + 1.
We know that x = n if and only if n x < n + 1.
So from n x < n + 1 we can infer that x = n = x

Answer & Explanation

Liberty Gates

Liberty Gates

Beginner2022-05-22Added 11 answers

Step 1
Go straight at it, but make careful and clear steps:
x = n n x < n + 1 n x > n 1
Step 2
x = n x = n n 1 < x n
Nicholas Cruz

Nicholas Cruz

Beginner2022-05-23Added 3 answers

Step 1
From the definitions, x = n n 1 < x n
and x = n n x < n + 1.
These are equivalent statements.
Step 2
Alternatively, WLOG the integer part is 0 (because the floor/ceiling commute with addition of an integer), and the fractional part is 0 or ( 0 , 1 ), say 0.5.
We have 0 = 0 = 0
and 0.5 = 1 = 0.5 .

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