- Thread starter
- #1

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?