- #1
oblixps
- 35
- 0
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, [math]lim_{n \to +\infty} liminf_{m \to +\infty} ||f_n - f_m|| = 0[/math]. 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?
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, [math]lim_{n \to +\infty} liminf_{m \to +\infty} ||f_n - f_m|| = 0[/math]. 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?