The sequence of functions {x,x/2,x/3,…}\{x,x/2,x/3,…\} is not uniformly convergent

I want to show the sequence of real functions (fn)(f_n) where fi(x)=xif_i(x)=\frac{x}{i} is not uniformly convergent, though it converges pointwise to f=0f=0.

Here’s my solution:

Let ε=1\varepsilon=1. Then given any NN, we can let n=N+1n=N+1 and x=N+2x=N+2. Then |xn|=|N+2N+1|>1|\frac{x}{n}|=\left|\frac{N+2}{N+1}\right|>1. Thus |fn−0|>ε|f_n-0|>\varepsilon and (fn)(f_n) is not uniformly convergent.

My question is whether I can let xx depend on nn like that, and whether what I did was valid.

Thanks!

=================

– Zachary Selk
2 days ago

Yes, forgot to mention that.
– ChrisWong
yesterday

=================