TUTORIAL 6
MTL503: REAL ANALYSIS
1. Suppose
lim fn (x) = f (x)
n→∞
forx ∈ E.
Define
Mn = sup |fn (x) − f (x)|.
x∈E
2.
3.
4.
5.
6.
7.
Then prove that fn converges to f uniformly if and only if
Mn → 0 as n → ∞.
Discuss the uniform convergence of
1
x ∈ (0, 1).
fn (x) =
1 + nx
Prove that there exists a continuous function on R which is
nowhere differentiable.
Prove that every uniformly convergent sequence of bounded
functions is uniformly bounded.
Let fn and gn are sequences of bounded functions converging
uniformly on E. Prove that fn gn converge uniformly.
Construct a sequence of functions fn and gn on E such that
they converge uniformly but fn gn does not converge uniformly
on E.
Define
x
fn (x) =
.
1 + nx2
Show that fn converge uniformly to a function f . Furthermore,
prove that the relation,
f ′ (x) = lim fn′ (x)
n→∞
hold for x ̸= 0 only.
1