Monday, 23 November 2015

real analysis - $||f||_pto ||f||_infty$ under general assumptions

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I am trying to show that for nonnegative $f$ on $\mathbb{R}$ if $||f||_1 $$ \lim_{p\uparrow \infty}||f||_p=||f||_\infty $$ I have tried to f...
Sunday, 22 November 2015

proof writing - For which natural numbers $n$ is $sqrt n$ irrational? How would you prove your answer?

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Can someone help me out here? I'm trying to prove this the same way as whether if $\sqrt 2$ is irrational, but i'm not sure what am ...
Saturday, 21 November 2015

proof verification - Prove by mathematical induction $4^n > n+1$

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Prove the following by mathematical induction: $4^n > n+1$ , for all integers $n ≥ 1$ Step 1: $n=1$ : LHS $= 4^{(1)} = 4 $ RHS $= (1) + 1...

calculus - Limit with unknown parameter

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What possible values can take $a,b\in\mathbb{R}$ such that $$ \lim\limits_{n\rightarrow\infty}\left( \sqrt[3]{an^{3}+bn^{2}+1}-\log _{5}\le...

Cauchy's functional equation -- additional condition

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Consider the function $f:R \to R$ $$f(x+y)=f(x)+f(y)$$ which is known as Cauchy's functional equation. I know that if $f$ is monotonic, ...
Friday, 20 November 2015

algebra precalculus - Bounds / Approximation to sum of squares of sum

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Can we define any tight upper / lower bound or approximation to the expression, $\sum_{i = 1}^{N}|x_{i} + y_{i}|^{2}$ in terms of $\sum_{i =...

real analysis - Find a one-to-one correspondence between $[0,1]$ and $(0,1)$.

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Establish a one-to-one correspondence between the closed interval $[0,1]$ and the open interval $(0,1)$. this is a problem in real analysi...