1. Prove: $1 + \frac12 x - \frac18 x^2 < \sqrt{1+x}, x > 0$

  2. Minkowski's triangle inequality for $0<p<1$

  3. How to prove that if $F$ belongs to $C^k$ then $F^{-1}$ also belongs to $C^k$?
  4. Proving this log function is continuous
  5. How to swap between $y$-fixed parts and $x$-fixed parts

  6. Proving not contraction mapping
  7. If $\omega=x dy - y dx$ is a $1$-form, how do I compute $d\omega$?

  8. Continuity of Functions from $[0,1]$ to $[0,1]$
  9. Complex differentiation - Final part

  10. Finding a domain with $C^1$ boundary
  11. $f$ is upper semi-continuous if $\{ x \in R^{n} : f(x) \geq \alpha\}$ is closed for all $\alpha \in R$

  12. Prove or disprove that $(b-a)^2(\log(b)-\log(a))^{-2}\ge ab$ for all $a,b>0$

  13. Check proof of contraction mapping

  14. Proving some statements, given that $|f(x)-f(y)| \leq M|x-y|^{\alpha}$
  15. How can I get integrating factor for $y(2x-y-1)dx+x(2y-x-1)dy=0$

  16. A characterization of equicontinuous sequence
  17. Calculate integral with help the Euler's integrals
  18. unable to draw regions in ipython notebook
  19. Show that $\frac{1}{3}<\int_0^1\frac{1}{1+x+x^2} \, dx <\frac{\pi}{4}$
  20. Disproving a misconception about Grandi's Sum

  21. What is the sum of the series with general term $\frac{(-1)^{n-1}\pi^{2n-1}}{(2n-1)!}$

  22. Arriving at a well-posed form for a variational problem

  23. $\lim\limits_{h\to 0} \frac{\sin(1/(a+h))-\sin(1/a)}{h}$
  24. Demonstrate that $(x+y)\ln \left(\frac{x+y}{2}\right) \leq x\ln x +y\ln y$

  25. Convergence of series $\sum_{n=1}^\infty \frac{1+x^{2n}}{n^6}$

  26. How to find closed form of the integral $\int_{-\infty}^{\infty}\frac{(c-x)e^{-a(b-x)^2}}{d + x^2}\,dx$?
  27. If $\lim\limits_{x \to \infty} f(f(x))= \infty$, $\lim\limits_{x \to \infty} f(x)=\pm \infty$
  28. I'm struggling to show it. Can anyone explaning how to show that answer?
  29. proving a mapping is contraction mapping
  30. $\sup\limits_{x\in X}\left(\sup\limits_{y\in y}|f(x,y)|\right) = \sup\limits_{y\in Y}\left(\sup\limits_{x\in X}|f(x,y)|\right)$
  31. Prove that $\lim_{h\to 0} \frac{g(a+h)-2g(a)+g(a-h)}{h^2} = g''(a)$
  32. Proof for inequality involving mean of $k$ closest values

  33. If $F$ is a closed subset of $[a,b]$ and length of $F$, $|F| = 0$ then is $F$ an empty set. True or False?

  34. How to determine the convergence or divergence of this sequence without resort to the L'Hospital's rule?

  35. Whether the indefinite integral of a differentiable function is differentiable or not

  36. If $f(x)$ is Riemann integrable on $[a,b]$ such that $\int_a^bf^2(x)dx=0$. Prove that $f(x)=0$ at every point of continuity within $[a,b]$.

  37. Examples of theorems which hold in Real numbers but not in Higher dimensions

  38. Convex sets. Real Analysis.

  39. Weak convergence (exercise)
  40. Evaluating $\sum\limits_{n=1}^\infty \frac{(-1)^{n+1}}{n}$

  41. F dense in E and their dual are the same

  42. Differentiability in the origin of $f(x,y)=\frac{x^2\sin{y^2}}{x^2+y^4}$

  43. Difficulties in stating mean value theorem for functions which are not continuous on a closed interval.
  44. Using Euler-Lagrange Equation

  45. $C^3$ boundary, locally the Levi form satisfies $L(r,t)\geq -C|r(z)||t|^2$ for tangent vectors $t$ and defining function $r$.
  46. Please convert to proper Math symbols the following

  47. Let $x_n=\frac{1}{3}+(\frac{2}{5})^2+(\frac{3}{7})^3+....(\frac{n}{2n+1})^2$ Is $(x_n) $cauchy sequence?

  48. Do we require a probability density function to be Borel-measurable?
  49. Proving $\lim\limits_{n\to \infty}\int\limits_{0}^{1} f_n(x)dx=0$

  50. Proving convexity of a $C^2$ function
  51. Proof that $\lim_{x\to \infty} (1+(x/n))^n = e^x$ Using Monotone Convergence Theorem
  52. Proof of the estimate $\;\| D^{2} u \|_{L^{p}}\sim \| \Delta u \|_{L^{p}}$ when $p>1$

  53. Is below subspace is connected?

  54. Evaluating a double integral using Fubini's Theorem

  55. Calculating the root of an almost polynomial expression.

  56. My Proof: Every convergent sequence is a Cauchy sequence.

  57. Let $g:\mathbb{R}\to \mathbb{R}$ satisfying $-1 \le g(x)\le 1$ for all $x \in \mathbb{R}$

  58. Norms on $\mathcal{P}_N$ Vector Space of Polynomials up to Order N

  59. find zeros of integral of nonlinear equations with variable bound

  60. Doubt in proof of Convex Function Continuous

  61. how to show that $\sum_{ n=1}^{∞ }a_n^+ = ∞ $?
  62. $\lim (x_n^k) = (\lim x_n)^k$

  63. Harmonic function, equivalence
  64. Not sure how to start proving that $\int_E f\,d\lambda$ = $\lim_{n \to \infty} \int_{E_n} f\,d\lambda$
  65. Definition of disconnected?

  66. Exponentials inequality: $|e^x + e^y| \leq |x - y|$ for $x,y<0$.
  67. Monotone convergence theorem implies Nested Interval property

  68. How to prove this specific function has limit either $\infty$ or 0?
  69. Showing that g is integrable and $\int^b_a{f}$ = $\int^b_a{g}$

  70. $\sum_{n=0}^{\infty} t^n \sum_{j=0}^{n} x^j Pr\{ Y_n=j\} = \frac{1-F(t)}{(1-t)(1-F(xt))} $

  71. proof $\lim x_n \le \lim y_n$ if $x_n \le y_n$ (clarification)
  72. Finding a function that solves the condition

  73. Chacking integrability of a piecewise function

  74. how to integrate $\sqrt{1-x^{2/3}}$

  75. Need help proving that $\int_{\bigcup_{n=1}^{\infty}E_n}f{\rm d}\lambda = \lim_{n\to\infty}\int_{E_n}f{\rm d}\lambda$
  76. When does $\int \limits_{\Bbb R} f \chi_{A} \,d\lambda$ = $\lambda$($A$) hold?
  77. If $f_n\to f$ in $L^p$ and $g_n$ is uniformly bounded and $g_n\to g$ a.e. then $g_nf_n\to gf$ in $L^p$.
  78. Let $f:[a,b]\to \mathbb{R}$ is integrable on $[a,b]$. Show that $f_1:[a,b]\to \mathbb{R}$ such that $f_1(x)=\sup\{f(x), 0\}$, is integrable.

  79. Help with Baby Rudin Chapter 8 Question 12
  80. Do any probability measure has to be countably additive by definition?
  81. For which (c,d) is the transformation linear?

  82. Real Analysis question that affects how to think about the Dirac delta function.

  83. How can I solve this limit?(From AMM)
  84. Prove that any polynomial function is continuous

  85. Proving that family of fractions is dense in $\mathbb{R}$
  86. Write trinomials as nonnegative linear combination

  87. Let $(f_n)$ be a sequence of measurable functions on a metric space.
  88. Inequality with sup and continuous function

  89. What is wrong with following reasoning?

  90. Confusion about little-o and sums
  91. Confusing derivative in $C([0,1])$
  92. First order characterization of strict convexity

  93. My Proof for subset of a finite set is finite
  94. Prove $\nabla^2 f(x) \preceq L I$ for convex $f$ with Lipschitz gradient

  95. Function converges iff when every sequence converges the sequence of function values converges

  96. two sequences of functions uniform convergence

  97. Derivative of a limit

  98. Prove by induction that $\frac{d^m }{dx^m}\Bigl(\frac{d^nf}{dx^n}\Bigl)=\frac{d^{m+n}f}{dx^{m+n}}$

  99. Show that function introduced by varying upper bound in Lebesgue integral is weakly differentiable

  100. Polynomials- Prove that function can be only polynomial