BMTC-133 Solved Assignment January 2026 | REAL ANALYSIS | IGNOU BSCG

Q: State whether the following statements are true or false. Justify your answers with a short proof or a counterexample

Answer available — download the full PDF for complete details.

Q: This question has multiple parts.

Answer available — download the full PDF for complete details.

Q: This question has multiple parts.

Answer available — download the full PDF for complete details.

Q: This question has multiple parts.

Answer available — download the full PDF for complete details.

Q: This question has multiple parts.

Answer available — download the full PDF for complete details.

Q: This question has multiple parts.

Answer available — download the full PDF for complete details.

Q1. State whether the following statements are true or false. Justify your answers with a short proof or a counterexample

i)) The negation of p V (~ q) is q → p. (100 words)
ii)) Q\N is countable. (100 words)
iii)) The equation x³ – 2x + 3 = 0 has a real root between -2 and 1. (100 words)
iv)) Every increasing sequence has a convergent subsequence. (100 words)
v)) The function defined as f(x) = { sin 2x / x, if x ≠ 0; 3, if x = 0 } has a removable discontinuity. (100 words)
vi)) An integrable function can have finitely many points of discontinuity. (100 words)
vii)) The series Σ(n=0 to ∞) (-1)^n / 2^n is divergent. (100 words)
viii)) The function f defined by f(x) = sinx(1 + cosx), x ∈ [0, 2π], has local maximum at x = π/2. (100 words)
ix)) Every constant function on [a, b] is Riemann Integrable. (100 words)
x)) The function f : R → R defined by f(x) = |x − 2| + |x − 1| is differentiable at x = 3. (100 words)

Key Points:

De Morgan's Laws are crucial for correct logical negation.
Subsets of countable sets are always countable (e.g., `Q\N`).
Intermediate Value Theorem guarantees a root for continuous functions with sign changes.
Bolzano-Weierstrass Theorem requires a sequence to be *bounded* to ensure a convergent subsequence.

Answer:

Q2. This question has multiple parts.

(a)) Using mathematical induction, prove that the sum of the squares of first 'n' natural numbers is n(n+1)(2n+1)/6. (200 words)
(b)) State Cauchy's second theorem on limits. Use it to show that: lim (n→∞) [(n + 1)(n+2)...(2n)]^(1/n) = e/4. (200 words)
(c)) Check whether or not the sequence (a_n)_(n∈N), defined by: a_(n+1) = 2 - 1/(a_n + 2) ∀n ≥ 1 and a₁ = 2 is convergent. (200 words)

Key Points:

Mathematical induction involves base case, inductive hypothesis, and inductive step.
Sum of squares of first n natural numbers is n(n+1)(2n+1)/6.
Cauchy's Second Theorem on Limits: If lim (a_(n+1)/a_n) = L, then lim (a_n)^(1/n) = L.
The limit of [(n + 1)...(2n)]^(1/n) using Cauchy's 2nd theorem is ∞, not e/4.

Answer: The provided question from BMTC-133 Real Analysis covers fundamental concepts including mathematical induction, properties of limits, specifically Cauchy's theorems, and the convergence of sequences using the Monotone Convergence Theorem. Each sub-question requires a thorough, step-by-step approach demonstrating proficiency in these core areas of real analysis. For mathematical induction, establishing the base case, inductive hypothesis, and inductive step is crucial. For limits, accurate applic...

Q3. This question has multiple parts.

(a)) Test the convergence of the series: Σ(n=1 to ∞) (-1)^(n+1)n / (n³+1). (150 words)
(b)) Find the radius of convergence of the series Σa_n x^n, where a_n = n! / n^n. (120 words)
(c)) Show that the set S = { (1+(-1)^n)/2 + 1/n : n ∈ N } is not closed. (120 words)

Key Points:

Alternating Series Test: Series Σ(-1)^(n+1)b_n converges if b_n > 0, decreasing, and lim b_n = 0.
Absolute convergence implies convergence; use comparison tests for absolute convergence.
Radius of convergence R for Σa_n x^n is R = 1 / (lim |a_(n+1)/a_n|) from the Ratio Test.
Euler's number 'e' is defined as the limit of (1 + 1/n)^n as n approaches infinity.

Answer: This response addresses three distinct problems from Real Analysis, covering series convergence, radius of convergence for power series, and properties of closed sets. It applies fundamental theorems and definitions typically taught in an IGNOU BMTC-133 course, such as the Alternating Series Test, the Ratio Test for power series, and the definition of limit points and closed sets. Each sub-question is tackled systematically, demonstrating the application of the relevant theoretical concepts to ...

Q4. This question has multiple parts.

(a)) Show that the function f : ]0, ∞[ → R defined as f(x) = 1/x is continuous on its domain, but not uniformly continuous there. Will the function be uniformly continuous on [c, ∞[ for any fixed c > 0? (250 words)
(b)) Examine the type of discontinuity of the function at x = 0 where f(x) = { |x|, if x ≠ 0; 2, if x = 0 }. (150 words)
(c)) Check whether the following sets are open or closed or neither: (i) {3n : n ∈ N} (ii) ]1, 6[ U [2, 8]. (200 words)

Key Points:

Continuity: Quotient of continuous functions is continuous where the denominator is non-zero.
Uniform Continuity: δ depends solely on ε, not on the specific point in the domain.
f(x)=1/x is continuous on ]0,∞[ but not uniformly continuous due to steepness near 0.
f(x)=1/x is uniformly continuous on [c,∞[ (c>0) because its derivative is bounded.

Answer:

Q5. This question has multiple parts.

(a)) Suppose that f and g are in R[a, b]. Then show that the function f + g is in R[a, b], and ∫(a to b) (f + g) = ∫(a to b) f + ∫(a to b) g. (200 words)
(b)) Discuss the convergence of the p-series Σ(n=1 to ∞) 1/(n^p), p > 0 by using the integral test. (200 words)

Key Points:

If f, g are Riemann integrable, f+g is also Riemann integrable on [a,b].
The integral of a sum equals the sum of integrals: ∫(f+g) = ∫f + ∫g.
Riemann integrability is shown by U(P, f+g) - L(P, f+g) < ε for any ε > 0.
Integral test requires f(x) to be positive, continuous, and decreasing.

Answer:

Q6. This question has multiple parts.

(a)) Prove that sin x < x, x ∈ [0, 1]. (200 words)
(b)) Prove that every convergent sequence is Cauchy. (120 words)
(c)) Disprove the statement: '(x + y)^n = x^n + y^n ∀n ∈ N, x,y ∈ Z' by providing a suitable counter-example. (100 words)

Key Points:

sin x < x proof uses f(x)=x-sin x, showing f(0)=0 and f'(x)=1-cos x > 0 for x ∈ (0,1].
Convergent sequence: terms get arbitrarily close to a limit L beyond a certain N.
Cauchy sequence: terms get arbitrarily close to each other beyond a certain N_0.
Convergent implies Cauchy proof uses triangle inequality |a_m-a_n| ≤ |a_m-L|+|L-a_n| and ε/2 argument.

Answer:

Q7. This question has multiple parts.

(a)) State Bolzano-Weierstrass theorem. Use it to check whether the set: S = { 1/m + 1/n : m,n ∈ N } has a limit point. (200 words)
(b)) Give one example for each of the following. Justify. (i) A set which is neither open or closed. (ii) A set which has no limit point. (100 words)
(c)) Check whether the set of integers is countable or not. (120 words)

Key Points:

Bolzano-Weierstrass Theorem: Every bounded infinite subset of R has at least one limit point.
Set S={1/m+1/n} is bounded (between 0 and 2) and infinite, so it has limit points (e.g., 0, 1/k).
Set (0, 1] is neither open (1 not interior) nor closed (0 is limit point, not in set).
The set of natural numbers N has no limit points, as no neighborhood contains infinite elements.

Answer:

Q8. This question has multiple parts.

(a)) Let f be a differentiable function whose derivative never vanishes on [a, b]. Show that f is either strictly increasing or strictly decreasing. (200 words)
(b)) If an infinite series Σa_n converges, then show that lim_(n→∞) a_n = 0. (200 words)

Key Points:

Darboux's Theorem: Differentiable function's derivative takes all intermediate values.
Constant Sign of Derivative: If f'(x) ≠ 0, its sign is constant on an interval.
Monotonicity from Derivative: f'(x)>0 implies strictly increasing; f'(x)<0 implies strictly decreasing.
Series Convergence: Defined by the convergence of its sequence of partial sums.

Answer:

BMTC-133 Solved Assignment — January 2026 / July 2026

REAL ANALYSIS | IGNOU Bachelor of Science (General) (CBCS) (BSCG)

BMTC-133—January 2026 / July 2026

REAL ANALYSIS

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BMTC-133 Assignment Questions — January 2026

Q1. State whether the following statements are true or false. Justify your answers with a short proof or a counterexample

Q2. This question has multiple parts.

Q3. This question has multiple parts.

Q4. This question has multiple parts.

Q5. This question has multiple parts.

Q6. This question has multiple parts.

Q7. This question has multiple parts.

Q8. This question has multiple parts.

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