Let X be a normed linear space (such as linear product space) and let {fn } n ϵ N be a sequence of elements of X.
We say that {fn }nϵN converges to fϵX and write fn↦f if lim//f−fn// =0
n→∞
∀ϵ>0 N>0 such that n>N→//f−fn//<ϵ
A function f:A→R where A⊂R
And suppose that CϵA . Then f is continuous at C if for every ϵ>0 there exist a δ>0 such that
∣x−c∣<δ and xϵA implies that ∣f(x)−f(c)∣<ϵ
fn:E→R continuous ∀n
fn→f uniformly.
Let ϵ>0,∃Nϵ such that ∀n>Nϵ
∣fn(x)−f(x)∣< 3ϵ ∀ xϵE
Fix x0ϵE which means ∃δ>0
Such that
∣x−x0∣ <δ tends to ∣fn(x)−fn(x0) ∣ < 3ϵ Is said to be continuous. We therefore show that ∣f(x)−f(x0)
=∣f(x)−fn(x)−fn(x0)+fn(x0)−f(x0)∣
≤ ∣f(x)−fn(x)∣+∣fn(x)−fn(x0)∣+∣fn(x0)−(fx0)∣
< 3ϵ + 3ϵ + 3ϵ = ϵ
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