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What does n tends to $\mathbf\infty$ mean ? Is it equivalent to saying $n>K$, $\;K\in \Bbb N$ ? For a Cauchy sequence, is $|a_m-a_n|<\varepsilon, \enspace m,n >K$ equivalent to saying $a_m-a_n \to 0$ as $m,n\to \infty$?

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For the first question, the answer is yes. When we say $n$ tends to $\infty $, we mean that for every natural number $N$, we may assume that $n>N$.

For the second question, the answer is also yes. Since Cauchy sequences are convergent both $a_m$ and $a_n$ converge to the same limit, therefore $ a_m-a_n$ approaches zero as $m$ and $n$ tend to infinity.

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The definition of a Cauchy sequence is this: if $(a_n)$ is a Cauchy sequence, then, for all $\varepsilon\in\mathbb{R}^{+}$ (for all positive real numbers $\varepsilon$), there exists some $K\in\mathbb{N}$ such that $|a_n-a_m|<\varepsilon$ for all $n,m>K$. In words, you can make the terms of $a_n$ get arbitrarily close together if you make their index in $(a_n)$ sufficiently large.

Saying some sequence $(x_n)$ tends to some value $k$ as $n$ tends to infinity is simply saying that the value of $x_n$ gets arbitrarily close to $k$ if you make $n$ big enough; you can make $x_n$ get as close to $k$ as you like, so $(x_n)$ tends to $k$. These example help illustrate how the term "tends to" simply means "gets arbitrarily close to". Similarly, saying $n$ tends to infinity means $n$ gets arbitrarily large, arbitrarily "close to" infinity.

The term "tends to" basically means"gets closer and closer to" such that you can make it as close to the value as you like if you go far enough through the sequence or close enough to a certain value in the function's domain, depending on the context.

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