MATH/MTHE 281 Midterm 2 Solutions — Full Questions, Answers & Detailed Proofs | Winter 2026 - Queens University.

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STUDENT NUMBER:

MATH/MTHE 281 — MIDTERM 2 — Mar. 2026

HAND IN
answers recorded
on exam paper




Instructors: Thomas Barthelmé & Neige Paulet


Question: 1 2 3 4 5 Total
Points: 5 5 14 13 13 50



ˆ No partial credit will be given for definitions, except in very rare cases.

ˆ Apart from the questions asking for a definition, all your answers need to be justified.

ˆ Calculators are not permitted.



This material is copyrighted and is for the sole use of students registered in Math/Mthe 281 and writing this
examination. This material shall not be distributed or disseminated. Failure to abide by these conditions is a
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STUDENT NUMBER:

P∞
1. [5 points] Let (an )∞
n=1 be a sequence of real numbers and S ∈ R. Give the definition of n=1 an =S
using ε, N and quantifiers (∀, ∃).


Solution: Let Sn = nk=1 ak be the sequence of partial sums. Then ∞
P P
n=1 an = S if and only if
for all ϵ > 0, there exists N such that, for all n > N , |S − Sn | < ϵ.
P∞
Equivalently,
Pn n=1 an = S if and only if for all ϵ > 0, there exists N such that, for all n > N ,
|S − k=1 ak | < ϵ.




2. [5 points] Define “x ∈ Rd is a boundary point of A.”


Solution: x is a boundary point of A if and only if for all r > 0, we have Br (x) ∩ A ̸= ∅ and
Br (x) ∩ A∁ ̸= ∅.