Definition 1. Let (X, d) be a metric space and let the xn be a sequence of points
of X. We say that a sequence converges to x0 ∈ X if and only if
lim d(xn , x0 ) = 0
n→∞
The point x0 is called a limit of the sequence and denoted by
lim xn = x0
n→∞
If the sequence has no limit we say it diverges.
Theorem 1. Let xn be a sequence in Rk and x0 ∈ Rk be its limit. Then
lim xn = x0 iff ∀i = 1, 2, ..., k xin = xi0
n
Theorem 2. Let (X, d) be a metric space. A set A ⊂ X is closed if and only
if every sequence in A that converges to a limit in X has its limit in A.
∞
Definition 2. A subsequence of (an )∞
n=1 is a sequence (bn )n=1 defined by bk =
ank , where n1 < n2 < n3 < ... is an increasing sequence of indices.
Theorem 3. If a sequence in a metric space converges then every subsequence
of that sequence converges to the same limit.
Definition 3. Let (X, d) be a metric space. A set A ⊂ X is called compact if
every sequence in A has a subsequence that converges to an element contained
in A.
Theorem 4. Let (X, d) be a metric space. If set A ⊂ X is compact, then it is
closed and bounded.
Theorem 5. A set A in an Euclidean space is compact if and only if it is closed
and bounded.
Definition 4. Let (X, d) be a metric space. The xn is said to be a Cauchy
sequence if and only if
∀ > 0 ∃N ∈ N ∀n, m ∈ N n, m > N d(xn , xm ) < .
Theorem 6. In a metric space any sequence that converges is a Cauchy sequence.
Definition 5. A complete metric space is a metric space in which every Cauchy
sequence is convergent.
Theorem 7. Euclidean space (Rn , dn ) for n ∈ N is a complete metric space.
Definition 6. Let (X, dX ), (Y, dY ) be metric spaces and let A ⊂ X. Then
y0 ∈ Y is the limit of the mapping f : A → Y in the limit point x0 ∈ A if and
only if
lim = y0 ,
xn →x0
for every sequence xn ∈ A.
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Definition 7. A mapping f : A → Y is said to be continuous in the point
x0 ∈ A if and only if the limit limxn →x0 f (xn ) exists and
lim f (xn ) = f (x0 ).
xn →x0
A mapping f : X → Y which is continuous in every point x ∈ X is said to be a
continuous mapping.
Theorem 8. Let (X, dX ) and (Y, dY ) be metric spaces. A mapping f : X → Y
is continuous if and only if
∀x0 ∈ X ∀ > 0 ∃δ > 0 ∀x ∈ X dX (x, x0 ) < δ ⇒ dY (f (x), f (x0 )) < Theorem 9. If f : X → Y and g : Y → Z are continuous, then so is the
composition g ◦ f : X → Z
Theorem 10. If f : X → Y is continuous and
• X is compact, then f (X) is compact
• X is connected, then f (X) is connected.
Theorem 11. Let (X, dX ) be a metric space, let A ⊂ X be a compact set and
let f : X → R be continuous. Then there exist two points m, M ∈ A such that
f (m) = inf{f (x) : x ∈ A}, f (M ) = sup{f (x) : x ∈ A}
Theorem 12. Let f : X → R be a continuous function and let A ⊂ X be a
connected set. Then for any two values a, b ∈ A and any y between f (a) and
f (b), there exists some c between a and b such that f (c) = y.
Norms
Definition 8. Function || · || : X → R is said to be a norm if and only if for
any x, y ∈ Rn and α ∈ R
• ||x||| > 0 and ||x|| = 0 if and only if x = 0
• ||αx|| = |α|||x||
• ||x + y|| ≤ ||x|| + ||y||.
An ordered pair (X, || · ||) we called a normed vector space.
Theorem 13. Let || · || : X → R be a norm. Then a function d : X × X → R
defined as
d(x, y) = ||x − y||
is a metric.
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