Uniform convergence implies uniform Cauchy: Let XXX be a set and let (fn)(f_n)(fn​) be functions fn:X→(Y,d)f_n:X\to (Y,d)fn​:X→(Y,d) into a metric space . If fn→ff_n\to ffn​→f uniformly on XXX, then (fn)(f_n)(fn​) is uniformly Cauchy : ∀ε>0 ∃N ∀m,n≥N: sup⁡x∈Xd(fn(x),fm(x))<ε. \forall\varepsilon>0\;\exists N\;\forall m,n\ge N:\ \sup_{x\in X} d(f_n(x),f_m(x))<\varepsilon. ∀ε>0∃N∀m,n≥N: supx∈X​d(fn​(x),fm​(x))<ε. ...

Uniform convergence of power series on compact subsets: Let ∑n=0∞an(x−x0)n \sum_{n=0}^\infty a_n (x-x_0)^n ∑n=0∞​an​(x−x0​)n be a power series with radius of convergence R>0R>0R>0. Then for every rrr with 0<r<R0<r<R0<r<R, the series converges uniformly on the closed set {x:∣x−x0∣≤r}. \{x:|x-x_0|\le r\}. {x:∣x−x0​∣≤r}. ...

Let (X,dX)(X,d_X)(X,dX​) be a metric space, let (Y,dY)(Y,d_Y)(Y,dY​) be a metric space, and let fn:E→Yf_n:E\to Yfn​:E→Y with E⊆XE\subseteq XE⊆X. The sequence (fn)(f_n)(fn​) converges uniformly on compact sets to f:E→Yf:E\to Yf:E→Y if for every compact set K⊆EK\subseteq EK⊆E (compact in the subspace metric), ...

Uniform convergence preserves boundedness: Let XXX be a set and let fn,f:X→Rf_n,f:X\to\mathbb{R}fn​,f:X→R (or into a normed space). If fn→ff_n\to ffn​→f uniformly on XXX and some fNf_NfN​ is bounded on XXX, then fff is bounded on XXX (and hence fnf_nfn​ is bounded for all sufficiently large nnn). ...

Corollary: Let (X,d)(X,d)(X,d) be a metric space and let fn:X→Rf_n:X\to\mathbb{R}fn​:X→R be continuous for all nnn. If fn→ff_n\to ffn​→f uniformly on XXX, then fff is continuous on XXX. Connection to parent theorem: This is exactly the uniform limit theorem for continuity, often restated as a corollary once the theorem has been proved. ...

Let fn:[a,b]→Rf_n:[a,b]\to\mathbb{R}fn​:[a,b]→R be Riemann integrable for each nnn and suppose fn→ff_n\to ffn​→f uniformly on [a,b][a,b][a,b]. Proposition: The limit function fff is Riemann integrable on [a,b][a,b][a,b]. The integrals converge to the integral of the limit: lim⁡n→∞∫abfn(x) dx=∫abf(x) dx. \lim_{n\to\infty}\int_a^b f_n(x)\,dx=\int_a^b f(x)\,dx. limn→∞​∫ab​fn​(x)dx=∫ab​f(x)dx. This is the standard mechanism for proving integrability of limits and justifies passing limits through integrals under uniform convergence. ...

Uniform limit theorem for continuity: Let (X,d)(X,d)(X,d) be a metric space and let (fn)(f_n)(fn​) be a sequence of continuous functions fn:X→Rf_n:X\to\mathbb{R}fn​:X→R (or into any metric space). If fn→ff_n\to ffn​→f uniformly on XXX, then fff is continuous on XXX. ...

Let XXX be a set and let F⊆RX\mathcal{F}\subseteq \mathbb{R}^XF⊆RX be a family of real-valued functions. The family F\mathcal{F}F is uniformly bounded if there exists M<∞M<\inftyM<∞ such that ∀f∈F ∀x∈X: ∣f(x)∣≤M. \forall f\in\mathcal{F}\ \forall x\in X:\ |f(x)|\le M. ∀f∈F ∀x∈X: ∣f(x)∣≤M. Equivalently, writing the sup norm ∥f∥∞=sup⁡x∈X∣f(x)∣\|f\|_\infty=\sup_{x\in X}|f(x)|∥f∥∞​=supx∈X​∣f(x)∣ (possibly +∞+\infty+∞), uniform boundedness is: sup⁡f∈F∥f∥∞<∞. \sup_{f\in\mathcal{F}} \|f\|_\infty <\infty. supf∈F​∥f∥∞​<∞. ...

The union of sets AAA and BBB is A∪B:={x:(x∈A) ∨ (x∈B)}.A\cup B := \{x : (x\in A)\ \lor\ (x\in B)\}.A∪B:={x:(x∈A) ∨ (x∈B)}. More generally, for an indexed family {Ai}i∈I\{A_i\}_{i\in I}{Ai​}i∈I​, the union is ⋃i∈IAi:={x:∃i∈I with x∈Ai}.\bigcup_{i\in I} A_i := \{x : \exists i\in I\ \text{with}\ x\in A_i\}.i∈I⋃​Ai​:={x:∃i∈I with x∈Ai​}.Unions are central in topology and analysis: open sets are closed under arbitrary unions, and coverings are families whose union contains the set of interest. ...

Uniqueness of limits: Let (X,d)(X,d)(X,d) be a metric space and let (xn)(x_n)(xn​) be a sequence in XXX. If xn→xx_n\to xxn​→x and xn→yx_n\to yxn​→y, then x=yx=yx=y. This lemma is a basic structural fact about metric convergence and is used everywhere (for example, to identify a limit by proving two different candidate limits). ...