Is it Cauchy?

Calculus Level 1

Consider the metric space consisting of continuous functions on [ 0 , 1 ] [0,1] with the metric d ( f , g ) = ∫ 0 1 ∣ f ( x ) − g ( x ) ∣ d x . d(f,g)=\int_0^1 |f(x)-g(x)|\, dx. Is the sequence f n ( x ) = x n f_n(x)=\frac xn a Cauchy sequence in this space?

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Samir Khan by Samir Khan , 23, USA

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1 solution

Samir Khan
Jul 11, 2016

Yes. Note that d ( f m , f n ) = ∫ 0 1 ∣ x m − x n ∣ d x = 1 2 ∣ 1 m − 1 n ∣ ≤ 1 2 ( 1 n + 1 m ) . d(f_m,f_n)=\int_0^1 \left|\frac{x}{m}-\frac{x}{n}\right|\, dx =\frac{1}{2}\left|\frac{1}{m}-\frac{1}{n}\right|\leq \frac{1}{2}\left(\frac{1}{n}+\frac{1}{m}\right). By making m , n m,n large enough, 1 / n , 1 / m 1/n, 1/m can be made arbitrarily small, so this sequence is Cauchy.

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