Do the following sequences of functions converge pointwise or uniformly?

Question

Determine whether the following functions converge pointless or uniformly on $[0, \infty)$.

1) $$ f_{n}(x) = \begin{cases} x - n & x \ge n \\ 0 & x < n \end{cases} $$ 2) $$ f_{n}(x) = \sqrt{x+\frac{1}{n}} - \sqrt{x} $$

I know that 1 converges pointwise to 0, but I am unsure whether it converges uniformly or not. I have no clue about the convergence of 2.

Answer 1

(1) does not converge uniformly. For every $n$, there exists $x \in [0,\rightarrow]$ with $f(x) = 1$. (Taking $x = n+1$) $[0,\rightarrow]$ means all real numbers greater than or equal to $0$.

For (2), notice that $\sqrt{x + (1/n)} \leq \sqrt{x} + \sqrt{1/n}$.