Question

Suppose we tried to apply our real analysis definitions/methods to the
set of rational numbers Q. In other words, in the definitions, we only
consider rational numbers. E.g., [0, 1] now means [0, 1] ∩ Q, etc. In
this setting:
(a) Find an open cover of [0, 1] that contains no finite subcover. Hint:
Fix an irrational number α ∈ [0, 1] (as a subset of the reals now!)
and for each (rational) q ∈ [0, 1] look for an open interval avoiding
α. (b) Prove that the function f(x) = 1
not bounded on [0, 2].
x2−2 is continuous on [0, 2] but

Suppose we tried to apply our real analysis definitions/methods to the set of rational numbers Q. In other words, in the definitions, we only consider rational numbers. E.g., [0, 1] now means [0, 1] n Q, etc. In this setting: (a) Find an open cover of [0, 1] that contains no finite subcover. Hint: Fix an irrational number a € [0, 1] (as a subset of the reals now! and for each (rational) qe [0, 1] look for an open interval avoiding a. 1 = (b) Prove that the function f(x) not bounded on (0, 2). is continuous on (0, 2] but

Answer 1

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