$n$ tending to infinity

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 $$m,n\to \infty$$?

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.
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.