Let X,, = U {k ;1;,k+ i) for any n E N. Prove that I] X,1 = Z by lling out the gaps in the #152 116 argument below. Proof. For any n E N and 1'...

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Let X,, = U {k —;1;,k+ i) for any n E N. Prove that I] X,1 = Z by ﬁlling out the gaps in the#152 116"argument below. Proof. For any n E N and 1' E R we have _< :1: <_. In particular, if :1: E Z, then :r E __ forany n E N as _. Therefore, E C X,1 for any n E N. As a result, _. We want to show that n X“ C Z. From the equivalence of the “if-then” statement and itsnEN contrapositive, it is enough to show that _. Notice that n Xn C IR as for all k E Z and allHEN n E N we have _. Therefore, we want to show that if 1: E_, then _. Assume r E_. Then,choose k E E such that k < r < k + 1. In particular, a.“ — k_ and (k +1)— :t_, so we canchoose or E N such that _ and _ because _. As a result, a >_ and a: <_. Therefore, 1: 5-3 _. So, :2: ¢_. Since _ and _, we have _. III