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Real Analysis
Show that {fn}, where fn : [0, 1] → R is the map fn(x) = x
n, is not uniformly convergent.
n, is not uniformly convergent.
Real Analysis
Show that {fn}, where fn : [0, 1] → R is the map fn(x) = x
n, is not uniformly convergent.
n, is not uniformly convergent.
Real Analysis
Let {vn} be a sequence in R2
, say vn = (xn, yn). Give R2
the || ||∞
norm. Show that limn→∞ vn → v if and only if limn→∞ xn = x and
limn→∞ yn = y where v = (x, y).
, say vn = (xn, yn). Give R2
the || ||∞
norm. Show that limn→∞ vn → v if and only if limn→∞ xn = x and
limn→∞ yn = y where v = (x, y).
Real Analysis
On R^n show that || . ||∞ ≤ || . ||2
.
Real Analysis
Under what condition the equality holds in |a+b|≤|a|+|b|,a,b∈R ?
Real Analysis
Suppose that {xn} is a convergent sequence and {yn} is such that for any ε>0, there exists M such that |xn−yn|< ϵ for all n≥M. Show that {yn} is a convergent sequence.
Real Analysis
Applying Bolzano-Weierstrass theorem show that the set
S={1+1/ n ∨ n∈N}∪{−1−1/ n ∨ n∈N}
must have a limit.
S={1+1/ n ∨ n∈N}∪{−1−1/ n ∨ n∈N}
must have a limit.
Real Analysis
Is every subsequence of a divergent sequence is divergent ? Justify.
Real Analysis
Prove that
a) Any Open ball is an open set
b) Any closed ball is closed set.
a) Any Open ball is an open set
b) Any closed ball is closed set.
Real Analysis
If {xn} does not converge to L, show that there exist > 0 and a subsequence {xnk
}
of {xn} such that |xnk − L| ≥ for each k ∈ N.
}
of {xn} such that |xnk − L| ≥ for each k ∈ N.
