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I learnt the converge definition in advanced calculus:

For any $epsilon>0$, there is a $N$, such that for all $n>N$, we have $|X_n-x|<epsilon$ then we say $X_n$ converge to $x$.

Then I learnt the consistent estimators in statistics:

For every $epsilon>0$ and every $theta in Theta$, $limlimits_ntoinftyP_theta(|W_n-theta| ge epsilon) = 0$

So my question is that is there any relationship between consistent estimators and converge sequence?

Are the two definition equivalent?

I believe the two definition must have something in common, but I can not find that.

And also , Is there any relationship between converge in distribution[$F_X_nto F_x$] and the definition of continuity in function? [$X_nto x_0, f(X_n) to f(x_0)]$

Perhaps this rephrasing will make the parallel more clear:

$X_nto X$ in probability $iff$ for any $epsilon>0$, there is an $N$ so $nge N$ implies $Eleft[fracX_n-Xright]<epsilon.$

For a proof, see convergence in probability induced by a metric.

In other words, if we define $d_P(X,Y)$ to be the funny quantity $Eleft[fracX-Yright]$, then $d_P(X_n,X)$ replaces $|X_n-X|$ in the usual definition of convergence. More generally, in any metric space with a distance function $d$, we have the following notion of convergence:

$x_nto x$ $iff$ for any $epsilon>0$, there is an $N$ so $nge N$ implies $d(x_n,x)<epsilon$.

Your calculus notion of convergence is the one from the metric space $mathbb R$ with distance function $d(x,y)=|x-y|$, while convergence in probability comes $d_P$.

Finally, there is a relationship between convergence and distribution and metric convergence. Let $X_n$ have cdf $F_n$, and let $X$ have cdf $F$.

$X_nto X$ in distribution $iff$ for all $x$ such that $F$ is continuous at $x$, we have $F_n(x)to F(x)$.

Also, convergence in distribution is induced by a metric in the same way convergence in probability is. In this case, the metric is
$$
d_textLevy(X,Y)=infepsilon>0:P(Xle x-epsilon)-epsilon<P(Yle x)le P(Xle x+epsilon)+epsilontext for all xin mathbb R
$$
See https://en.wikipedia.org/wiki/L%C3%A9vy_metric.