Let an=n1 . Let ϵ>0 is given. Then there exist some N>0 satisfying N>ϵ21.( Archimedean property of real numbers)
Therefore ∀ n≥N,
∣an−0∣=∣n1∣=n1≤N1<ϵ .
Hence from the definition of limits an→0.
Let bn=n(−1)n. Then ∀n≥N,
∣bn−0∣=∣n(−1)n∣=n1≤N1<ϵ .
Hence from the definition of limits bn→0.
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