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Tests For Uniform Convergence : Theorem 1 (Mₙ _ Test) :                  Let <fₙ> be a sequence of function defined on a metric space X .      Let  lim  fₙ(x) = f(x)   ∀ x ∈ X and let            n-->∞          Mₙ = Sup {|fₙ(x) - f(x)| : x∈ X}   Then <fₙ> converges uniformly to f iff     Mₙ-->0 as n-->∞. Proof Of Necessary Part :          Let us suppose the sequence <fₙ> of functions converges uniformly to f on X . Then by definition , for a given ε > 0 ∃ a positive integer m (independent of x)     such that  n≥ m ⇒|fₙ(x) - f(x)| < ε ∀ x∈X    Also , Mₙ is the supremum of |fₙ(x) - f(x)|. Therefore |fₙ(x) - f(x)| < ε ∀ n≥m  ∀ x∈X         ⇒ Mₙ = Sup |fₙ(x) - f(x)| < ε...

Uniform Convergence :        Uniform Convergence is a part of real analysis which is discussed in detail below. Definition :             Let (X,d) be a metric space and f be a function from X to R . Also for each n ∈ ℕ let fₙ : X ---> R . Then , the sequence of functions <fₙ> converges pointwise to the function f , if for each x ∈ X , the sequence of real numbers < fₙ(x) > converges to the real number f(x) .                   Therefore <fₙ(x) > converges pointwise to f if  lim   fₙ(x) = f(x)    ∀ x ∈ X .                              n-->∞ For Example :            Let < fₙ > be the sequence defined by fₙ  : R --> R such that fₙ(x) = x / n  ∀ x∈R , n∈N  . Show that the sequence conver...

Examples Related To Riemann Integral :  Example 1 :                                                                                                1                  Show that ∫ x⁴ dx = 1 / 5                                     0  Proof :                     Let us consider the partition P in the interval [0,1] as { 0,1/n, 2/n , 3/n .....n/n }.  Since  f(x) = x⁴ , so the supremum and infimum of the function in the interval is (i/n)⁴ and (i-1 / n) ⁴  .    Length of the interval = Δxᵢ = 1-0...

Mean Value Theorem Of Integrability :   First Mean Value Theorem :          If a function f is continuous on [a,b] then ∃ a number ξ in [a,b] such that          b         ∫ f dx = f(ξ) (b - a)         a f is continuous , therefore f ∈ R on [ a,b ] . Proof :          Given that function f is continuous on  [ a, b] . Let m , M be the infimum and supremum of f in [ a,b ] . Then clearly       we have                                      b                 m( b - a ) ≤ ∫ f dx ≤ M( b - a )                                     a So , ∃ a number μ ∈ [ m, M ] such that    ...

Fundamental Theorems Of Integral Calculus : First Fundamental Theorem Of Integral Calculus : Theorem 1 :                 If a function f is bounded and integrable on [ a,b ] , then the function F defined as                    x                            F(x) = ∫ f(t) dt , a≤x≤b                                         0 is continuous on [ a,b ] and further more , if f is continuous at a point of [ a,b ] , then F is derivable at c and F'(c) = f(c) . Proof :           It is given that the function f is bounded . Then by definition ∃ a number k such that |f(x)|≤ k for x ∈ [ a,b ] ......(1)       Let x₁ , x₂ ∈[ a,b ] such that a≤x₁≤x₂≤b .    ...

Darboux's Theorem For Integrability :           If  f is a bounded function on [a,b] , then to every ε > 0 , there corresponds         δ > 0, such that                         -b  (A)   U(P,f) < ∫ f dx + ε                           a                                                          b   (B)    L(P,f) > ∫ f dx - ε                        - a for every partition P of [ a,b ] with norm μ(P) < δ . Proof : (A)        Given f is bounded function of [a,b] , so there exists a positive number k...

Conditions Of Integrability :    ( A ) First Form Of Integrability :                   A necessary and sufficient condition for the integrability of a bounded function f is that to every ε > 0 , there corresponds δ > 0 such that for every partition P of [a,b] with norms μ(P) < δ ,   U(P,f) - L(P,f) < ε . Proof Of Necessary Part :               Let's suppose that the bounded function f be integrable i.e          b          - b            b        ∫ f dx = ∫ f dx = ∫ f dx        -a            a             a      Let ε > 0 be a positive small number . By Darboux's Theorem ∃ a δ > 0 such that for every partition P with norm μ(P) < δ we...