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confusion in proof involving lim and liminf


May 20, 2012
In a proof showing that the vector space of all functions from a metric space X to the complex numbers C is complete under the supremum norm, there was a line near the end that i was confused about.

starting here:
\(sup|f - fn| \leq liminf_{m \to +\infty} ||f_n - f_m|| \)

the next step then is to take the limit as n approaches infinite on both sides.

Since we are assuming that {f_n} is Cauchy I know that the limit as m, n approach infinite of ||fn - fm|| is 0.

However, the very next line says that the right hand side is 0. so in other words, \(\displaystyle lim_{n \to +\infty} liminf_{m \to +\infty} ||f_n - f_m|| = 0\). i feel that it is a bit of a jump from the previous line and it isn't entirely obvious to me why this line is true.

Could someone help fill in the missing step or two?


Feb 1, 2012
Do you mean the vector space of bounded functions on this metric space?