# Uniform Convergence. Real Analysis

Could some please suggest me with a step by step method to proving a sequence of functions is uniformly convergent. I don't quite understand the difference between pointwise and uniform convergence either.

An example of a sequence of real-valued continuous functions converging uniformly to a real-valued continuous function with steps would be greatly appreciated!

Thank you very much! :)

• The most standard example is $f_n(x) = x^n$. It converges pointwisely on $[0,1]$ to the function $f(x) = \begin{cases} 1 & x=1 \\ 0 & \text{otherwise}\end{cases}$ but the convergence is uniform. – user99914 May 6 '17 at 8:12
• Uniform convergence means that if you fix a tube around your target function, the eventually the future functions in your sequence will each be in that tube. Tube == epsilon of allowed error from the value of the target function at each time. eventually ... each be in == there is some N so that for all n >= N... draw some tubes. – Lorenzo May 6 '17 at 8:30
• @JohnMa is not uniform ... (just a typo, but maybe confusing) – Lorenzo May 6 '17 at 8:31
• Thanks for the help! Sorry is the above example uniformly convergent? Would anyone suggest a step by step method/criterion to check if a sequence of functions are uniformly convergent – USERMATHS May 6 '17 at 8:37
• Sorry I can't use latex but could someone please help provide a systematic approach to show why f_n: [0,1] -> R defined by f_n(x)=(n^2)x(1-x)^n converges pointwise to zero function but does not converge uniformly. Thank you! – USERMATHS May 6 '17 at 8:53