# Function Convergence Proofs

** Please see the attached file for the complete problem description **

Let f be a function defined on R and, for each natural number n, define the function f_n by....

Decide whether or not you believe the statement is true.

#### Solution Preview

... |f_n(x) - f(x)|<e for any n>N and any x in R. Hence f_n(x) converges to f(x) uniformly on R.

(b) False

We consider f(x) = x^2. Then for e=1>0, for any n>0, we can set x = n and we have

|f_n(x) - f(x)| = |f(x + 1/n) - f(x)| = |f(n + 1/n) - f(n)| = (n + 1/n)^2 - n^2

= n^2 + 2 + 1/n^2 - n ^2 = 2 + 1/n^2 > 2 = e

Thus f_n(x) does not converge uniformly on R.

(c) True

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