Cauchy criterion for convergence
If a sequence converges, it is the same as saying it matches the Cauchy criterion for convergence.
Contents
Cauchy Sequence
A sequence [math](a_n)^\infty_{n=1}[/math] is Cauchy if:
[math]\forall\epsilon>0\exists N\in\mathbb{N}:n> m> N\implies d(a_m,a_n)<\epsilon[/math]
Theorem
A sequence converges if and only if it is Cauchy
TODO: proof, easy stuff
Interesting examples
[math]f_n(t)=t^n\rightarrow 0[/math] in [math]\|\cdot\|_{L^1}[/math]
Using the [math]\|\cdot\|_{L^1}[/math] norm stated here for convenience: [math]\|f\|_{L^p}=\left(\int^1_0|f(x)|^pdx\right)^\frac{1}{p}[/math] so [math]\|f\|_{L^1}=\int^1_0|f(x)|dx[/math]
We see that [math]\|f_n\|_{L^1}=\int^1_0x^ndx=\left[\frac{1}{n+1}x^{n+1}\right]^1_0=\frac{1}{n+1}[/math]
This clearly [math]\rightarrow 0[/math] - this is [math]0:[0,1]\rightarrow\mathbb{R}[/math] which of course has norm [ilmath]0[/ilmath], we think of this from the sequence [math](\|f_n-0\|_{L^1})^\infty_{n=1}\rightarrow 0\iff f_n\rightarrow 0[/math]
Functional Analysis
