Prove that ⋃ n ∈ N ( − n , 1 n ] = (...

uplakanimkk

uplakanimkk

Answered

2022-07-07

Prove that n N ( n , 1 n ] = ( , 1 ]
I understand that for this to be true, I have to show that n N ( n , 1 n ] ( , 1 ] and ( , 1 ] n N ( n , 1 n ] , however, I'm stuck on figuring it out. I think the latter can be proven by contradiction somehow.

Answer & Explanation

Tanner Hamilton

Tanner Hamilton

Expert

2022-07-08Added 12 answers

Step 1
Show mutual inclusion.
First suppose that x n N ( n , 1 n ] . Choose n N such that x ( n , 1 n ] . Since x 1 n 1, x ( , 1 ]. This shows that n N ( n , 1 n ] ( , 1 ].
Step 2
Conversely, suppose that x ( , 1 ]. If 0 x 1, we may choose n N such that x 1 n . Then x ( n , 1 n ], so x n N ( n , 1 n ] . On the other hand, if x < 0, there exists n N such that x > n. x < 0 < 1 n , so x ( n , 1 n ], so x n N ( n , 1 n ] . This shows that n N ( n , 1 n ] ( , 1 ].
Desirae Washington

Desirae Washington

Expert

2022-07-09Added 5 answers

Step 1
x ( n , 1 n ] n N   s.t. x ( n , 1 n ]
as n N and as n > then x ( , 1 ]
Step 2
For the second, we know that for every number a there exists number b > a such that b N , in the same way it works backwards, so that for every a there exists b < a such that b Z .
Step 3
If 0 x 1 we take n = 1, else we take n = y where y is a smaller number than x that is in N and satisfies that y < x, so we get that again x ( y , 1 y ].

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