Hello!

I am working on what seems to be a very "concrete" exercise in showing convergence.

on the interval $\displaystyle [0,1]$, show that the functions converge to the function f(x) point-wise, and determine if they converge uniformly.

a) $\displaystyle \{f_n(x)\}_{n=1}^{\infty} = \{\frac{2x}{1+nx}\}$ where $\displaystyle f(x) = 0$

b) $\displaystyle \{f_n(x)\}_{n=1}^{\infty} = \{\frac{cos(nx)}{\sqrt{n}}\}$ where $\displaystyle f(x) = 0$

c) $\displaystyle \{f_n(x)\}_{n=1}^{\infty} = \{\frac{n^3x}{1+n^4x}\}$ where $\displaystyle f(x) = 0$

What I've got so far...

So, to show a function is convergent point-wise, we just try to show that the NUMBER sequence $\displaystyle \{f_n(x)\} $ converges to $\displaystyle f(x) = 0$ so, $\displaystyle \forall \epsilon > 0, \exists N $ such that $\displaystyle n > N$ implies $\displaystyle f_n(x) - 0 < \epsilon$

Here is where I get confused. Do we define an epsilon in terms of n?! Which would depend on the series of functions in all 3 cases?

And for uniform convergence, we are seeking a similar result but with an epsilon which is independent of n?

Any help/guidance to set up the question further appreciated!!