# Tagged Questions

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### Consecutive natural numbers [duplicate]

Please I want to know what is the most appropriate expression that if it is asked to find the counterexample of "The product of any three consecutive natural numbers is divisible by 9" My expression ...
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### show that $f(x,y) =2x^2 + 3y$ is differentiable at $(0,0)$ by producing a linear function

Here's the question: Prove that $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ defined $f(x,y) = 2x^2 + 3y$ is differentiable at $\begin{bmatrix} 0\\0 \end{bmatrix}$ by producing a linear function T and ...
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### Prove by induction $1 + \frac{1}{2} + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{2^n} \ge 1 + \frac {n}{2}$

Prove by induction $1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \ldots + \frac{1}{2^n} \ge 1 + \frac {n}{2}$ I can't explain in words how the left hand side of the equation is achieved soI shall ...
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### Using Darboux Sums to Prove Upper and Lower Integrals

Define $f:[0, 1]\rightarrow\mathbb{R}$ as f (x) \equiv \left\{\begin{array}{l l} x & \text{if } x\in [0, 1]\cap \mathbb{Q}\\ 0& \text{if }x\in [0, ...
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### Proving a language is not recognizable

I have the following question that I just want to verify I have done correctly. Let $L$, $L_1$, $L_2$ $\subseteq \Sigma^*$ such that $L = L_1 \cup L_2$, and $L_2$ is decidable. Prove that if $L$ is ...
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### the boundary of set in euclidean space is closed.

MY Attempt: Let $\partial A$ be the boundary of any set $A \subseteq \mathbb{R}^n$. We show $R^n \setminus \partial A$ is open. Pick $x \in \mathbb{R}^n \subseteq \partial A$. Then by definition, we ...
### Every Cauchy sequence in a metric space $(X,d)$ is bounded.
MY attempt: Suppose $(x_n)$ is a Cauchy sequence in $(X,d)$. Take $\varepsilon = 1$. Hence, can find $N$ such that $d(x_m,x_n) < 1$ for all $n,m > N$. Also, we have $d(x_N, x_n) < 1$ ...