# Math Snap

## Given $\lim \left(x_{n}\right)=x$. One of the following sentences is FALSE. Select one: a. None of them b. $(\forall \epsilon>0)(\exists K=K(\epsilon) \in \mathbb{N})\left[n \geq K \Rightarrow\left|x_{n}-x\right|<\epsilon\right]$ c. The sequence $\left(x_{n}\right)$ converges to $x$ d. $(\forall \epsilon>0)(\exists K=K(\epsilon) \in \mathbb{N})\left[n \geq K \Rightarrow x-\epsilon e. $(\forall \in>0)(\exists K=K(\epsilon) \in \mathbb{N})\left[n \geq K \Rightarrow x_{n} \in V_{\epsilon}(x)\right]$

#### STEP 1

Assumptions 1. We are given that $\lim \left(x_{n}\right)=x$. 2. We need to identify which of the given statements is FALSE. 3. The statements involve the definition of the limit of a sequence and its properties.

#### STEP 2

Recall the definition of the limit of a sequence. A sequence $(x_n)$ converges to $x$ if for every $\epsilon > 0$, there exists a natural number $K$ such that for all $n \geq K$, $|x_n - x| < \epsilon$.

#### STEP 3

Examine statement (b): $(\forall \epsilon>0)(\exists K=K(\epsilon) \in \mathbb{N})\left[n \geq K \Rightarrow\left|x_{n}-x\right|<\epsilon\right]$ This is the formal definition of the limit of a sequence. Therefore, statement (b) is TRUE.

#### STEP 4

Examine statement (c): "The sequence $(x_{n})$ converges to $x$" This is a restatement of the given information $\lim \left(x_{n}\right)=x$. Therefore, statement (c) is TRUE.

#### STEP 5

Examine statement (d): $(\forall \epsilon>0)(\exists K=K(\epsilon) \in \mathbb{N})\left[n \geq K \Rightarrow x-\epsilon This statement is equivalent to saying that $|x_n - x| < \epsilon$, which is the definition of the limit of a sequence. Therefore, statement (d) is TRUE.

#### STEP 6

Examine statement (e): $(\forall \epsilon>0)(\exists K=K(\epsilon) \in \mathbb{N})\left[n \geq K \Rightarrow x_{n} \in V_{\epsilon}(x)\right]$ Here, $V_{\epsilon}(x)$ denotes the $\epsilon$-neighborhood of $x$, which means $x_n$ is within $\epsilon$ distance from $x$. This is equivalent to $|x_n - x| < \epsilon$. Therefore, statement (e) is TRUE.

##### SOLUTION

Since statements (b), (c), (d), and (e) are all TRUE, the only possible answer is: a. None of them Therefore, the FALSE statement is: a. None of them
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