Prove that $\lim_{n \to \infty} \frac {1}{\sqrt n}=0$.
Attempy: $\forall \varepsilon>0 $, we have to find $M\in N$ such that $|\frac {1}{\sqrt n}-0|<\varepsilon$ for $n \ge M$.
Let $\varepsilon > \frac 1{\sqrt M}$. We can do this since $M \in N$, and note that $n\ge M \rightarrow \sqrt n \ge \sqrt M \rightarrow 1/\sqrt n \le 1/\sqrt M.$
Then, for $n \ge M$, we have that $|\frac {1}{\sqrt n}-0| = |\frac {1}{\sqrt n}| = \frac {1}{\sqrt n}$ (because $n\ge M \in N) = \frac 1{\sqrt M} <\varepsilon$.
Therefore, by definition of convergence, $\lim_{n \to \infty} \frac {1}{\sqrt n}=0$.
This is an assignment question, and marking criteria is quite strict. So, could you pick any minor mistake?
Thank you in advance.