Can every element in the arbitrary space be converged to?

Question

When I have for example $\mathbb R$ then I'm able to create a sequence which will converge to any of the elements in $\mathbb R$:

\begin{align} \frac{1}{n} &\rightarrow 0\\ \frac{1}{n} + 1 &\rightarrow 1\\ \end{align}

But, is this possible in every space? Or they exist some special elements in some spaces, which are "non-convergeable", while other elements in the same space can be converged to with suitable sequence?

Answer 1

Yes, because (whenever convergence makes sense) for every element $x$ in the space, the constant sequence $x$ converges to $x$